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Properties

Label 4-416000-1.1-c1e2-0-17
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 2·9-s + 13-s + 25-s − 12·29-s + 12·37-s − 12·41-s − 2·45-s − 14·49-s − 12·53-s − 4·61-s − 65-s + 12·73-s − 5·81-s − 12·89-s − 12·97-s − 12·101-s + 8·109-s − 24·113-s + 2·117-s + 2·121-s − 125-s + 127-s + 131-s + 137-s + 139-s + 12·145-s + ⋯
L(s)  = 1  − 0.447·5-s + 2/3·9-s + 0.277·13-s + 1/5·25-s − 2.22·29-s + 1.97·37-s − 1.87·41-s − 0.298·45-s − 2·49-s − 1.64·53-s − 0.512·61-s − 0.124·65-s + 1.40·73-s − 5/9·81-s − 1.27·89-s − 1.21·97-s − 1.19·101-s + 0.766·109-s − 2.25·113-s + 0.184·117-s + 2/11·121-s − 0.0894·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.996·145-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: \(1\)
Selberg data: \((4,\ 416000,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
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$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.7.a_o
11$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.11.a_ac
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.a_ac
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.19.a_ac
23$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.23.a_o
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.29.m_dq
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.31.a_ac
37$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.37.am_dq
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.41.m_eo
43$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.43.a_o
47$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.47.a_o
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.53.m_fm
59$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \) 2.59.a_bu
61$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.61.e_as
67$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.67.a_ak
71$C_2^2$ \( 1 + 94 T^{2} + p^{2} T^{4} \) 2.71.a_dq
73$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.73.am_gk
79$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \) 2.79.a_bu
83$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \) 2.83.a_di
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.89.m_ig
97$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.97.m_ig
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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

−8.221213711239283060884920394495, −7.929057798166127189915522240778, −7.63472455479138020982609129766, −6.93444732862684193941643263625, −6.66263734642032794158802456088, −6.12581621869779989172759848600, −5.50599002454060796510383309833, −5.07591698136251992484349942385, −4.40960313757204426277642793908, −4.05341903334901116797367023231, −3.40324912116407497185599711142, −2.92820321529991941023296261529, −1.93041709804939199534979027593, −1.37680542094331207279052849785, 0, 1.37680542094331207279052849785, 1.93041709804939199534979027593, 2.92820321529991941023296261529, 3.40324912116407497185599711142, 4.05341903334901116797367023231, 4.40960313757204426277642793908, 5.07591698136251992484349942385, 5.50599002454060796510383309833, 6.12581621869779989172759848600, 6.66263734642032794158802456088, 6.93444732862684193941643263625, 7.63472455479138020982609129766, 7.929057798166127189915522240778, 8.221213711239283060884920394495

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