Properties

Label 4-1148e2-1.1-c1e2-0-8
Degree $4$
Conductor $1317904$
Sign $1$
Analytic cond. $84.0307$
Root an. cond. $3.02767$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 3·5-s + 2·7-s + 4·9-s + 2·11-s − 10·13-s + 9·15-s − 2·17-s − 3·19-s − 6·21-s − 3·23-s − 6·27-s + 17·29-s + 11·31-s − 6·33-s − 6·35-s + 6·37-s + 30·39-s − 2·41-s − 16·43-s − 12·45-s − 18·47-s + 3·49-s + 6·51-s − 9·53-s − 6·55-s + 9·57-s + ⋯
L(s)  = 1  − 1.73·3-s − 1.34·5-s + 0.755·7-s + 4/3·9-s + 0.603·11-s − 2.77·13-s + 2.32·15-s − 0.485·17-s − 0.688·19-s − 1.30·21-s − 0.625·23-s − 1.15·27-s + 3.15·29-s + 1.97·31-s − 1.04·33-s − 1.01·35-s + 0.986·37-s + 4.80·39-s − 0.312·41-s − 2.43·43-s − 1.78·45-s − 2.62·47-s + 3/7·49-s + 0.840·51-s − 1.23·53-s − 0.809·55-s + 1.19·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1317904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1317904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1317904\)    =    \(2^{4} \cdot 7^{2} \cdot 41^{2}\)
Sign: $1$
Analytic conductor: \(84.0307\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1148} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 1317904,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
7$C_1$ \( ( 1 - T )^{2} \)
41$C_1$ \( ( 1 + T )^{2} \)
good3$C_4$ \( 1 + p T + 5 T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
5$D_{4}$ \( 1 + 3 T + 9 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
17$D_{4}$ \( 1 + 2 T - p T^{2} + 2 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 3 T + 11 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 3 T + 45 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 17 T + 127 T^{2} - 17 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 11 T + 63 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 6 T + 70 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 16 T + 137 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 9 T + 123 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - T + 89 T^{2} - p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 6 T + 79 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 7 T + 117 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 10 T + 115 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
79$D_{4}$ \( 1 + 26 T + 314 T^{2} + 26 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 49 T^{2} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - T + 97 T^{2} - p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 7 T + 47 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
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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

−9.961414740221143141082937553408, −9.359829169732999118224323952834, −8.458997331607138900203396001902, −8.282082728411612588929065157092, −8.024447531732714624305872337895, −7.48978747266291864979269984048, −6.98833163798159531286378604341, −6.66691879071017716711945135472, −6.25276846102323624839260353989, −5.91878841737183448085647024331, −4.95062669236222872955198003075, −4.84879565949978601910816944378, −4.53582088453010019548437907418, −4.36822042737295187210412247150, −3.34860006362171274484112762775, −2.79560697896910721361933140368, −2.07949825066371185577319425321, −1.22097015233537491478469959868, 0, 0, 1.22097015233537491478469959868, 2.07949825066371185577319425321, 2.79560697896910721361933140368, 3.34860006362171274484112762775, 4.36822042737295187210412247150, 4.53582088453010019548437907418, 4.84879565949978601910816944378, 4.95062669236222872955198003075, 5.91878841737183448085647024331, 6.25276846102323624839260353989, 6.66691879071017716711945135472, 6.98833163798159531286378604341, 7.48978747266291864979269984048, 8.024447531732714624305872337895, 8.282082728411612588929065157092, 8.458997331607138900203396001902, 9.359829169732999118224323952834, 9.961414740221143141082937553408

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