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Properties

Label 4-756e2-1.1-c1e2-0-24
Degree $4$
Conductor $571536$
Sign $-1$
Analytic cond. $36.4416$
Root an. cond. $2.45696$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 4-s − 3·8-s − 16-s + 25-s + 4·29-s + 5·32-s − 6·37-s − 7·49-s + 50-s − 16·53-s + 4·58-s + 7·64-s − 6·74-s − 7·98-s − 100-s − 16·106-s − 18·109-s + 32·113-s − 4·116-s + 3·121-s + 127-s − 3·128-s + 131-s + 137-s + 139-s + 6·148-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 1.06·8-s − 1/4·16-s + 1/5·25-s + 0.742·29-s + 0.883·32-s − 0.986·37-s − 49-s + 0.141·50-s − 2.19·53-s + 0.525·58-s + 7/8·64-s − 0.697·74-s − 0.707·98-s − 0.0999·100-s − 1.55·106-s − 1.72·109-s + 3.01·113-s − 0.371·116-s + 3/11·121-s + 0.0887·127-s − 0.265·128-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.493·148-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571536 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(571536\)    =    \(2^{4} \cdot 3^{6} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(36.4416\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 571536,\ (\ :1/2, 1/2),\ -1)\)

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)$Isogeny Class over $\mathbf{F}_p$
bad2$C_2$ \( 1 - T + p T^{2} \)
3 \( 1 \)
7$C_2$ \( 1 + p T^{2} \)
good5$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \) 2.5.a_ab
11$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.11.a_ad
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.13.a_k
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.17.a_c
19$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.19.a_bd
23$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.23.a_bt
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.29.ae_ck
31$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.31.a_cb
37$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.37.g_df
41$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \) 2.41.a_ab
43$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.43.a_by
47$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.47.a_cg
53$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.53.q_go
59$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.59.a_de
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.a_w
67$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.67.a_ack
71$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.71.a_dp
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.73.a_aeg
79$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.79.a_cg
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.83.a_fa
89$C_2^2$ \( 1 - 169 T^{2} + p^{2} T^{4} \) 2.89.a_agn
97$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.97.a_aby
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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

−8.279493202876599174540976743759, −7.80871439650669563623703314290, −7.28988512181529862656460178022, −6.66737192106165416393420731241, −6.33584110929613012711273555844, −5.89641260043268682832070404287, −5.27362695173636192465981288537, −4.88278867616077249543717168838, −4.52724739684782657322229284403, −3.89840574712728579273575554830, −3.31842795816732221448883047733, −2.96303881813488564738966701271, −2.12790397741908633100235640460, −1.21988870350823421143905956764, 0, 1.21988870350823421143905956764, 2.12790397741908633100235640460, 2.96303881813488564738966701271, 3.31842795816732221448883047733, 3.89840574712728579273575554830, 4.52724739684782657322229284403, 4.88278867616077249543717168838, 5.27362695173636192465981288537, 5.89641260043268682832070404287, 6.33584110929613012711273555844, 6.66737192106165416393420731241, 7.28988512181529862656460178022, 7.80871439650669563623703314290, 8.279493202876599174540976743759

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