Properties

Degree $4$
Conductor $13176900$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 2·3-s + 3·4-s − 2·5-s + 4·6-s − 4·7-s + 4·8-s + 3·9-s − 4·10-s + 6·12-s − 2·13-s − 8·14-s − 4·15-s + 5·16-s + 6·18-s − 14·19-s − 6·20-s − 8·21-s − 10·23-s + 8·24-s + 3·25-s − 4·26-s + 4·27-s − 12·28-s − 10·29-s − 8·30-s + 2·31-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.15·3-s + 3/2·4-s − 0.894·5-s + 1.63·6-s − 1.51·7-s + 1.41·8-s + 9-s − 1.26·10-s + 1.73·12-s − 0.554·13-s − 2.13·14-s − 1.03·15-s + 5/4·16-s + 1.41·18-s − 3.21·19-s − 1.34·20-s − 1.74·21-s − 2.08·23-s + 1.63·24-s + 3/5·25-s − 0.784·26-s + 0.769·27-s − 2.26·28-s − 1.85·29-s − 1.46·30-s + 0.359·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13176900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13176900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(13176900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 11^{4}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{3630} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 13176900,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
3$C_1$ \( ( 1 - T )^{2} \)
5$C_1$ \( ( 1 + T )^{2} \)
11 \( 1 \)
good7$D_{4}$ \( 1 + 4 T + 15 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 2 T + 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 31 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
23$D_{4}$ \( 1 + 10 T + 68 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
31$D_{4}$ \( 1 - 2 T + 36 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 4 T + 51 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 2 T + 56 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 6 T + 68 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 4 T + 50 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 2 T + 104 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 6 T + 52 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 6 T - 16 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 16 T + 186 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 2 T + 131 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 2 T + 72 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 26 T + 323 T^{2} + 26 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 12 T + 202 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 14 T + 240 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.248390750755783289191677225462, −7.987380278764521617704016378298, −7.51653426498253516174255243545, −7.28752323474667045110842603472, −6.62767653326035805056188088007, −6.58460240435678906851513840738, −6.08310664686536769519996331058, −6.01112544186951641746073845838, −5.08712811974075621937109589836, −4.85414793444078759597647877069, −4.32802134742402898192968693657, −3.90179327274581027441527375593, −3.68594827446393547320995992960, −3.56200413906123759278691398321, −2.77793729437577847741261455881, −2.55476549388887777358395752590, −1.89630619936456074985842817717, −1.74979085625501629176136000626, 0, 0, 1.74979085625501629176136000626, 1.89630619936456074985842817717, 2.55476549388887777358395752590, 2.77793729437577847741261455881, 3.56200413906123759278691398321, 3.68594827446393547320995992960, 3.90179327274581027441527375593, 4.32802134742402898192968693657, 4.85414793444078759597647877069, 5.08712811974075621937109589836, 6.01112544186951641746073845838, 6.08310664686536769519996331058, 6.58460240435678906851513840738, 6.62767653326035805056188088007, 7.28752323474667045110842603472, 7.51653426498253516174255243545, 7.987380278764521617704016378298, 8.248390750755783289191677225462

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