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Properties

Label 4-1040e2-1.1-c1e2-0-1
Degree $4$
Conductor $1081600$
Sign $1$
Analytic cond. $68.9637$
Root an. cond. $2.88174$
Motivic weight $1$
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  + 2·5-s − 6·9-s − 4·13-s − 16·17-s − 25-s + 8·29-s − 14·37-s + 2·41-s − 12·45-s − 20·61-s − 8·65-s + 10·73-s + 27·81-s − 32·85-s + 6·89-s − 26·97-s + 14·109-s + 24·117-s − 12·125-s + 127-s + 131-s + 137-s + 139-s + 16·145-s + 149-s + 151-s + 96·153-s + ⋯
L(s)  = 1  + 0.894·5-s − 2·9-s − 1.10·13-s − 3.88·17-s − 1/5·25-s + 1.48·29-s − 2.30·37-s + 0.312·41-s − 1.78·45-s − 2.56·61-s − 0.992·65-s + 1.17·73-s + 3·81-s − 3.47·85-s + 0.635·89-s − 2.63·97-s + 1.34·109-s + 2.21·117-s − 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.32·145-s + 0.0819·149-s + 0.0813·151-s + 7.76·153-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1081600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1081600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1081600\)    =    \(2^{8} \cdot 5^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(68.9637\)
Root analytic conductor: \(2.88174\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1081600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4490988304\)
\(L(\frac12)\) \(\approx\) \(0.4490988304\)
\(L(\frac{3}{2})\) 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)$Isogeny Class over $\mathbf{F}_p$
bad2 \( 1 \)
5$C_2$ \( 1 - 2 T + p T^{2} \)
13$C_2$ \( 1 + 4 T + p T^{2} \)
good3$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.3.a_g
7$C_2^2$ \( 1 + p^{2} T^{4} \) 2.7.a_a
11$C_2^2$ \( 1 + p^{2} T^{4} \) 2.11.a_a
17$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.17.q_du
19$C_2^2$ \( 1 + p^{2} T^{4} \) 2.19.a_a
23$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.23.a_abu
29$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.29.ai_cw
31$C_2^2$ \( 1 + p^{2} T^{4} \) 2.31.a_a
37$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.37.o_du
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.41.ac_c
43$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.43.a_adi
47$C_2^2$ \( 1 + p^{2} T^{4} \) 2.47.a_a
53$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.53.a_adm
59$C_2^2$ \( 1 + p^{2} T^{4} \) 2.59.a_a
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.61.u_io
67$C_2^2$ \( 1 + p^{2} T^{4} \) 2.67.a_a
71$C_2^2$ \( 1 + p^{2} T^{4} \) 2.71.a_a
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.73.ak_by
79$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.79.a_agc
83$C_2^2$ \( 1 + p^{2} T^{4} \) 2.83.a_a
89$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.89.ag_s
97$C_2$ \( ( 1 + 8 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) 2.97.ba_na
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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

−10.44039244843377352654454675472, −9.476516544105126344103563058933, −9.279835545062348133498515382213, −8.884951082925717907167398268543, −8.622677576114694553389534395651, −8.261188005445901927353752317042, −7.70112347581210492532644965741, −6.95686763136530186518201924952, −6.75558948013166677782297201206, −6.24686510682329106805704949532, −6.06481485900211638932677946746, −5.18467227085501603303214734170, −5.15684659029971979157326938900, −4.48965490345573500260159749505, −4.09071203977580949899166563998, −3.05008745980644672242972066717, −2.81967322886742369071081063043, −2.07598852842877490144855559905, −1.98609898200997055386994284383, −0.27300611148814900051635790492, 0.27300611148814900051635790492, 1.98609898200997055386994284383, 2.07598852842877490144855559905, 2.81967322886742369071081063043, 3.05008745980644672242972066717, 4.09071203977580949899166563998, 4.48965490345573500260159749505, 5.15684659029971979157326938900, 5.18467227085501603303214734170, 6.06481485900211638932677946746, 6.24686510682329106805704949532, 6.75558948013166677782297201206, 6.95686763136530186518201924952, 7.70112347581210492532644965741, 8.261188005445901927353752317042, 8.622677576114694553389534395651, 8.884951082925717907167398268543, 9.279835545062348133498515382213, 9.476516544105126344103563058933, 10.44039244843377352654454675472

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