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Properties

Label 4-416000-1.1-c1e2-0-10
Degree $4$
Conductor $416000$
Sign $1$
Analytic cond. $26.5245$
Root an. cond. $2.26940$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 2·9-s + 13-s + 12·17-s + 25-s + 12·41-s + 2·45-s − 2·49-s + 8·61-s + 65-s − 5·81-s + 12·85-s − 24·97-s + 12·101-s − 4·109-s − 24·113-s + 2·117-s − 10·121-s + 125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 24·153-s + 157-s + ⋯
L(s)  = 1  + 0.447·5-s + 2/3·9-s + 0.277·13-s + 2.91·17-s + 1/5·25-s + 1.87·41-s + 0.298·45-s − 2/7·49-s + 1.02·61-s + 0.124·65-s − 5/9·81-s + 1.30·85-s − 2.43·97-s + 1.19·101-s − 0.383·109-s − 2.25·113-s + 0.184·117-s − 0.909·121-s + 0.0894·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 1.94·153-s + 0.0798·157-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(416000\)    =    \(2^{8} \cdot 5^{3} \cdot 13\)
Sign: $1$
Analytic conductor: \(26.5245\)
Root analytic conductor: \(2.26940\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 416000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.587443063\)
\(L(\frac12)\) \(\approx\) \(2.587443063\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 - T \)
13$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 2 T + p T^{2} ) \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.3.a_ac
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.7.a_c
11$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.11.a_k
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.17.am_cs
19$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.19.a_ao
23$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.23.a_c
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.a_w
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.31.a_bu
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.37.a_aba
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.41.am_eo
43$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.43.a_abi
47$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.47.a_by
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.a_cs
59$C_2^2$ \( 1 - 62 T^{2} + p^{2} T^{4} \) 2.59.a_ack
61$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.61.ai_dy
67$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.67.a_aw
71$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.71.a_aba
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.73.a_fm
79$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.79.a_aby
83$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.83.a_by
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.89.a_fm
97$C_2$$\times$$C_2$ \( ( 1 + 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.97.y_mw
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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

−8.604692423085697030113367024018, −8.084139645673546791107696329021, −7.64093925642942581113036575506, −7.42651388767149809812654477623, −6.77739661101318683549481994897, −6.25191400384433061780792991457, −5.75492192729522238563419641466, −5.38453595017614536002849237565, −4.94904685216138818883573518698, −4.08751520953072361631922672489, −3.79116949942168468660348079005, −3.05446559990037517845742775821, −2.53166033072789949587316265410, −1.49321135254807399133247796395, −1.02115982770191657919040928459, 1.02115982770191657919040928459, 1.49321135254807399133247796395, 2.53166033072789949587316265410, 3.05446559990037517845742775821, 3.79116949942168468660348079005, 4.08751520953072361631922672489, 4.94904685216138818883573518698, 5.38453595017614536002849237565, 5.75492192729522238563419641466, 6.25191400384433061780792991457, 6.77739661101318683549481994897, 7.42651388767149809812654477623, 7.64093925642942581113036575506, 8.084139645673546791107696329021, 8.604692423085697030113367024018

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