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Properties

Label 4-416000-1.1-c1e2-0-9
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 + 4·9-s − 13-s + 4·17-s + 25-s − 4·29-s + 8·37-s + 18·41-s − 4·45-s + 6·49-s + 8·53-s − 16·61-s + 65-s − 20·73-s + 7·81-s − 4·85-s − 12·89-s + 34·97-s − 24·101-s − 8·109-s + 12·113-s − 4·117-s + 2·121-s − 125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 0.447·5-s + 4/3·9-s − 0.277·13-s + 0.970·17-s + 1/5·25-s − 0.742·29-s + 1.31·37-s + 2.81·41-s − 0.596·45-s + 6/7·49-s + 1.09·53-s − 2.04·61-s + 0.124·65-s − 2.34·73-s + 7/9·81-s − 0.433·85-s − 1.27·89-s + 3.45·97-s − 2.38·101-s − 0.766·109-s + 1.12·113-s − 0.369·117-s + 2/11·121-s − 0.0894·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-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

\(L(1)\) \(\approx\) \(1.981789425\)
\(L(\frac12)\) \(\approx\) \(1.981789425\)
\(L(\frac{3}{2})\) 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$ \( ( 1 + T )( 1 + p T^{2} ) \)
good3$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.3.a_ae
7$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.7.a_ag
11$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.11.a_ac
17$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.17.ae_w
19$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.19.a_ao
23$C_2^2$ \( 1 + 44 T^{2} + p^{2} T^{4} \) 2.23.a_bs
29$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.29.e_cg
31$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.31.a_ag
37$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.37.ai_di
41$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.41.as_fy
43$C_2^2$ \( 1 + 56 T^{2} + p^{2} T^{4} \) 2.43.a_ce
47$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.47.a_ag
53$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.53.ai_eo
59$C_2^2$ \( 1 + 42 T^{2} + p^{2} T^{4} \) 2.59.a_bq
61$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.q_ha
67$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \) 2.67.a_ade
71$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.71.a_k
73$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.73.u_jm
79$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.79.a_abe
83$C_2^2$ \( 1 + 82 T^{2} + p^{2} T^{4} \) 2.83.a_de
89$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) 2.89.m_cs
97$C_2$$\times$$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 - 16 T + p T^{2} ) \) 2.97.abi_so
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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

−8.726140085369102134134905114219, −7.909915744810047199386742250035, −7.64698373881530702036637805092, −7.39377005025037627054628419655, −6.98207177646388163868616811871, −6.22171820635899872779014658110, −5.84941736918724986788498331331, −5.38398920293026995557106869763, −4.53425149374061104155571317386, −4.31440237075864746955125686399, −3.86466410836183472185599727060, −3.07043885090940006969892979930, −2.51126912895381452494807330502, −1.59046004128422166491975098224, −0.833990460541444771394976155637, 0.833990460541444771394976155637, 1.59046004128422166491975098224, 2.51126912895381452494807330502, 3.07043885090940006969892979930, 3.86466410836183472185599727060, 4.31440237075864746955125686399, 4.53425149374061104155571317386, 5.38398920293026995557106869763, 5.84941736918724986788498331331, 6.22171820635899872779014658110, 6.98207177646388163868616811871, 7.39377005025037627054628419655, 7.64698373881530702036637805092, 7.909915744810047199386742250035, 8.726140085369102134134905114219

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