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Properties

Label 8-51e4-1.1-c2e4-0-1
Degree $8$
Conductor $6765201$
Sign $1$
Analytic cond. $3.72924$
Root an. cond. $1.17883$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s + 12·3-s + 8·4-s + 8·5-s − 48·6-s − 4·7-s − 20·8-s + 90·9-s − 32·10-s − 16·11-s + 96·12-s + 16·14-s + 96·15-s + 47·16-s − 24·17-s − 360·18-s − 40·19-s + 64·20-s − 48·21-s + 64·22-s − 28·23-s − 240·24-s + 48·25-s + 540·27-s − 32·28-s − 384·30-s + 12·31-s + ⋯
L(s)  = 1  − 2·2-s + 4·3-s + 2·4-s + 8/5·5-s − 8·6-s − 4/7·7-s − 5/2·8-s + 10·9-s − 3.19·10-s − 1.45·11-s + 8·12-s + 8/7·14-s + 32/5·15-s + 2.93·16-s − 1.41·17-s − 20·18-s − 2.10·19-s + 16/5·20-s − 2.28·21-s + 2.90·22-s − 1.21·23-s − 10·24-s + 1.91·25-s + 20·27-s − 8/7·28-s − 12.7·30-s + 0.387·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6765201 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6765201 ^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(6765201\)    =    \(3^{4} \cdot 17^{4}\)
Sign: $1$
Analytic conductor: \(3.72924\)
Root analytic conductor: \(1.17883\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 6765201,\ (\ :1, 1, 1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.036039931\)
\(L(\frac12)\) \(\approx\) \(2.036039931\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_1$ \( ( 1 - p T )^{4} \)
17$C_2^2$ \( 1 + 24 T + 16 p T^{2} + 24 p^{2} T^{3} + p^{4} T^{4} \)
good2$D_4\times C_2$ \( 1 + p^{2} T + p^{3} T^{2} + 5 p^{2} T^{3} + 49 T^{4} + 5 p^{4} T^{5} + p^{7} T^{6} + p^{8} T^{7} + p^{8} T^{8} \)
5$D_4\times C_2$ \( 1 - 8 T + 16 T^{2} + 8 p^{2} T^{3} - 1472 T^{4} + 8 p^{4} T^{5} + 16 p^{4} T^{6} - 8 p^{6} T^{7} + p^{8} T^{8} \)
7$D_4\times C_2$ \( 1 + 4 T + 76 T^{2} + 92 p T^{3} + 88 p^{2} T^{4} + 92 p^{3} T^{5} + 76 p^{4} T^{6} + 4 p^{6} T^{7} + p^{8} T^{8} \)
11$D_4\times C_2$ \( 1 + 16 T + 162 T^{2} + 2320 T^{3} + 25154 T^{4} + 2320 p^{2} T^{5} + 162 p^{4} T^{6} + 16 p^{6} T^{7} + p^{8} T^{8} \)
13$D_4\times C_2$ \( 1 - 372 T^{2} + 87110 T^{4} - 372 p^{4} T^{6} + p^{8} T^{8} \)
19$D_4\times C_2$ \( 1 + 40 T + 800 T^{2} + 21800 T^{3} + 560194 T^{4} + 21800 p^{2} T^{5} + 800 p^{4} T^{6} + 40 p^{6} T^{7} + p^{8} T^{8} \)
23$D_4\times C_2$ \( 1 + 28 T + 228 T^{2} - 19660 T^{3} - 549576 T^{4} - 19660 p^{2} T^{5} + 228 p^{4} T^{6} + 28 p^{6} T^{7} + p^{8} T^{8} \)
29$D_4\times C_2$ \( 1 + 1352 T^{2} - 17160 T^{3} + 913952 T^{4} - 17160 p^{2} T^{5} + 1352 p^{4} T^{6} + p^{8} T^{8} \)
31$D_4\times C_2$ \( 1 - 12 T + 1188 T^{2} - 51348 T^{3} + 1155960 T^{4} - 51348 p^{2} T^{5} + 1188 p^{4} T^{6} - 12 p^{6} T^{7} + p^{8} T^{8} \)
37$D_4\times C_2$ \( 1 - 24 T + 272 T^{2} - 12744 T^{3} + 269120 T^{4} - 12744 p^{2} T^{5} + 272 p^{4} T^{6} - 24 p^{6} T^{7} + p^{8} T^{8} \)
41$D_4\times C_2$ \( 1 - 16 T + 786 T^{2} - 94096 T^{3} + 1629602 T^{4} - 94096 p^{2} T^{5} + 786 p^{4} T^{6} - 16 p^{6} T^{7} + p^{8} T^{8} \)
43$D_4\times C_2$ \( 1 - 28 T + 392 T^{2} - 40964 T^{3} + 4131742 T^{4} - 40964 p^{2} T^{5} + 392 p^{4} T^{6} - 28 p^{6} T^{7} + p^{8} T^{8} \)
47$D_{4}$ \( ( 1 + 28 T - 794 T^{2} + 28 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
53$D_4\times C_2$ \( 1 + 72 T + 2592 T^{2} + 68904 T^{3} - 1598206 T^{4} + 68904 p^{2} T^{5} + 2592 p^{4} T^{6} + 72 p^{6} T^{7} + p^{8} T^{8} \)
59$D_4\times C_2$ \( 1 - 300 T + 45000 T^{2} - 4408500 T^{3} + 306132254 T^{4} - 4408500 p^{2} T^{5} + 45000 p^{4} T^{6} - 300 p^{6} T^{7} + p^{8} T^{8} \)
61$D_4\times C_2$ \( 1 + 216 T + 29336 T^{2} + 2714496 T^{3} + 192126752 T^{4} + 2714496 p^{2} T^{5} + 29336 p^{4} T^{6} + 216 p^{6} T^{7} + p^{8} T^{8} \)
67$D_{4}$ \( ( 1 - 116 T + 12244 T^{2} - 116 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
71$D_4\times C_2$ \( 1 + 132 T + 4388 T^{2} - 702516 T^{3} - 83662408 T^{4} - 702516 p^{2} T^{5} + 4388 p^{4} T^{6} + 132 p^{6} T^{7} + p^{8} T^{8} \)
73$D_4\times C_2$ \( 1 + 88 T + 2034 T^{2} - 330968 T^{3} - 27815518 T^{4} - 330968 p^{2} T^{5} + 2034 p^{4} T^{6} + 88 p^{6} T^{7} + p^{8} T^{8} \)
79$D_4\times C_2$ \( 1 - 44 T + 3684 T^{2} + 696844 T^{3} - 30070408 T^{4} + 696844 p^{2} T^{5} + 3684 p^{4} T^{6} - 44 p^{6} T^{7} + p^{8} T^{8} \)
83$D_4\times C_2$ \( 1 + 36 T + 648 T^{2} - 181764 T^{3} - 91531042 T^{4} - 181764 p^{2} T^{5} + 648 p^{4} T^{6} + 36 p^{6} T^{7} + p^{8} T^{8} \)
89$D_{4}$ \( ( 1 + 88 T + 15856 T^{2} + 88 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 - 204 T + 10406 T^{2} + 1901796 T^{3} - 333907198 T^{4} + 1901796 p^{2} T^{5} + 10406 p^{4} T^{6} - 204 p^{6} T^{7} + p^{8} T^{8} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.01867430014103034062461235954, −10.54626439088105382124338973463, −10.29512773636664260458137983242, −10.09838757738391669152664503304, −9.970874168332234679461280013152, −9.426106252819499558786836994678, −9.295932904765400561907478762577, −9.253705226481295356346724668221, −8.839041365003803972783453105905, −8.423393128994751554548433046469, −8.292387135014624493932584584703, −8.051114475537324158620778949930, −7.942718891727660796504234703804, −7.04292406561326305324180060921, −7.00067946542818283164254415571, −6.43056542900883699146836423130, −6.39806154868790916249655567979, −5.52003392197586351918941495472, −4.51116015937718616918351570435, −4.38070225784177123072000317374, −3.58336072826324776490056380321, −2.84047574483842761701039483923, −2.67582156082064280631503319361, −2.17769821720340409618728490492, −1.82297097560309954529971281420, 1.82297097560309954529971281420, 2.17769821720340409618728490492, 2.67582156082064280631503319361, 2.84047574483842761701039483923, 3.58336072826324776490056380321, 4.38070225784177123072000317374, 4.51116015937718616918351570435, 5.52003392197586351918941495472, 6.39806154868790916249655567979, 6.43056542900883699146836423130, 7.00067946542818283164254415571, 7.04292406561326305324180060921, 7.942718891727660796504234703804, 8.051114475537324158620778949930, 8.292387135014624493932584584703, 8.423393128994751554548433046469, 8.839041365003803972783453105905, 9.253705226481295356346724668221, 9.295932904765400561907478762577, 9.426106252819499558786836994678, 9.970874168332234679461280013152, 10.09838757738391669152664503304, 10.29512773636664260458137983242, 10.54626439088105382124338973463, 11.01867430014103034062461235954

Graph of the $Z$-function along the critical line