Properties

Label 8-136e4-1.1-c1e4-0-0
Degree $8$
Conductor $342102016$
Sign $1$
Analytic cond. $1.39079$
Root an. cond. $1.04209$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·3-s + 4·5-s − 4·7-s + 10·9-s + 4·11-s − 16·15-s + 4·17-s − 12·19-s + 16·21-s + 4·23-s + 10·25-s − 20·27-s − 12·29-s + 12·31-s − 16·33-s − 16·35-s + 4·37-s − 12·41-s + 12·43-s + 40·45-s − 6·49-s − 16·51-s + 20·53-s + 16·55-s + 48·57-s + 28·59-s − 28·61-s + ⋯
L(s)  = 1  − 2.30·3-s + 1.78·5-s − 1.51·7-s + 10/3·9-s + 1.20·11-s − 4.13·15-s + 0.970·17-s − 2.75·19-s + 3.49·21-s + 0.834·23-s + 2·25-s − 3.84·27-s − 2.22·29-s + 2.15·31-s − 2.78·33-s − 2.70·35-s + 0.657·37-s − 1.87·41-s + 1.82·43-s + 5.96·45-s − 6/7·49-s − 2.24·51-s + 2.74·53-s + 2.15·55-s + 6.35·57-s + 3.64·59-s − 3.58·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 17^{4}\)
Sign: $1$
Analytic conductor: \(1.39079\)
Root analytic conductor: \(1.04209\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{136} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{12} \cdot 17^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.6597216632\)
\(L(\frac12)\) \(\approx\) \(0.6597216632\)
\(L(\frac{3}{2})\) 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)$
bad2 \( 1 \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
good3$C_2^2$$\times$$C_2^2$ \( ( 1 - 2 T^{2} + p^{2} T^{4} )( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} ) \)
5$D_4\times C_2$ \( 1 - 4 T + 6 T^{2} - 4 T^{3} + 2 T^{4} - 4 p T^{5} + 6 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
7$D_4\times C_2$ \( 1 + 4 T + 22 T^{2} + 68 T^{3} + 226 T^{4} + 68 p T^{5} + 22 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
11$D_4\times C_2$ \( 1 - 4 T + 6 T^{2} - 4 T^{3} + 2 T^{4} - 4 p T^{5} + 6 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2$ \( ( 1 - p T^{2} )^{4} \)
19$D_4\times C_2$ \( 1 + 12 T + 72 T^{2} + 396 T^{3} + 1982 T^{4} + 396 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
23$D_4\times C_2$ \( 1 - 4 T + 22 T^{2} + 4 p T^{3} - 18 p T^{4} + 4 p^{2} T^{5} + 22 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
29$D_4\times C_2$ \( 1 + 12 T + 54 T^{2} + 108 T^{3} + 162 T^{4} + 108 p T^{5} + 54 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
31$D_4\times C_2$ \( 1 - 12 T + 54 T^{2} - 108 T^{3} + 162 T^{4} - 108 p T^{5} + 54 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
37$D_4\times C_2$ \( 1 - 4 T + 6 T^{2} - 4 T^{3} + 2 T^{4} - 4 p T^{5} + 6 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
41$D_4\times C_2$ \( 1 + 12 T + 134 T^{2} + 1132 T^{3} + 8450 T^{4} + 1132 p T^{5} + 134 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
43$D_4\times C_2$ \( 1 - 12 T + 72 T^{2} - 300 T^{3} + 926 T^{4} - 300 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
47$C_4\times C_2$ \( 1 + 28 T^{2} + 1414 T^{4} + 28 p^{2} T^{6} + p^{4} T^{8} \)
53$D_4\times C_2$ \( 1 - 20 T + 200 T^{2} - 1740 T^{3} + 13982 T^{4} - 1740 p T^{5} + 200 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \)
59$D_4\times C_2$ \( 1 - 28 T + 392 T^{2} - 4284 T^{3} + 37982 T^{4} - 4284 p T^{5} + 392 p^{2} T^{6} - 28 p^{3} T^{7} + p^{4} T^{8} \)
61$D_4\times C_2$ \( 1 + 28 T + 438 T^{2} + 4892 T^{3} + 43170 T^{4} + 4892 p T^{5} + 438 p^{2} T^{6} + 28 p^{3} T^{7} + p^{4} T^{8} \)
67$D_{4}$ \( ( 1 - 8 T + 118 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
71$D_4\times C_2$ \( 1 - 12 T + 134 T^{2} - 1612 T^{3} + 14210 T^{4} - 1612 p T^{5} + 134 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
73$D_4\times C_2$ \( 1 - 4 T + 102 T^{2} + 124 T^{3} + 3138 T^{4} + 124 p T^{5} + 102 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
79$D_4\times C_2$ \( 1 - 36 T + 662 T^{2} - 8388 T^{3} + 83042 T^{4} - 8388 p T^{5} + 662 p^{2} T^{6} - 36 p^{3} T^{7} + p^{4} T^{8} \)
83$D_4\times C_2$ \( 1 + 12 T + 72 T^{2} + 780 T^{3} + 8126 T^{4} + 780 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
89$D_4\times C_2$ \( 1 - 260 T^{2} + 30694 T^{4} - 260 p^{2} T^{6} + p^{4} T^{8} \)
97$D_4\times C_2$ \( 1 + 28 T + 214 T^{2} - 1828 T^{3} - 42398 T^{4} - 1828 p T^{5} + 214 p^{2} T^{6} + 28 p^{3} T^{7} + p^{4} T^{8} \)
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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

−9.903092684985055607655007149584, −9.500405471843425007995337275833, −9.449042012786102215074637020806, −9.094579766826229471778090474118, −8.762924745979933267427006587083, −8.451742630397633008855426735450, −7.991922040003695595499183319479, −7.69990800604350471163860286566, −7.14122774594751221110520638400, −6.71979159039829428066635544096, −6.53538641281901950419130297654, −6.53172030298454632147526814749, −6.47303129405001764875303400762, −5.91186569801768685588346898984, −5.51855675954712629303288004227, −5.50894171060124303790079343854, −5.14431452819139227153559737165, −4.64608879087912381497830631296, −4.22183942878266038100927665634, −3.78770171404982152271895317798, −3.65744991360970000038252171712, −2.69085940400383342724202037154, −2.29727338820350504257485595163, −1.65159012694999088176465622880, −0.844277900644065299776300647648, 0.844277900644065299776300647648, 1.65159012694999088176465622880, 2.29727338820350504257485595163, 2.69085940400383342724202037154, 3.65744991360970000038252171712, 3.78770171404982152271895317798, 4.22183942878266038100927665634, 4.64608879087912381497830631296, 5.14431452819139227153559737165, 5.50894171060124303790079343854, 5.51855675954712629303288004227, 5.91186569801768685588346898984, 6.47303129405001764875303400762, 6.53172030298454632147526814749, 6.53538641281901950419130297654, 6.71979159039829428066635544096, 7.14122774594751221110520638400, 7.69990800604350471163860286566, 7.991922040003695595499183319479, 8.451742630397633008855426735450, 8.762924745979933267427006587083, 9.094579766826229471778090474118, 9.449042012786102215074637020806, 9.500405471843425007995337275833, 9.903092684985055607655007149584

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