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Properties

Label 4-416000-1.1-c1e2-0-11
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 − 5·9-s − 3·13-s + 11·17-s + 25-s + 5·29-s − 7·37-s − 15·41-s + 5·45-s + 11·49-s − 24·53-s + 3·65-s + 18·73-s + 16·81-s − 11·85-s + 14·89-s + 2·97-s − 7·101-s + 4·109-s + 113-s + 15·117-s + 3·121-s − 125-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  − 0.447·5-s − 5/3·9-s − 0.832·13-s + 2.66·17-s + 1/5·25-s + 0.928·29-s − 1.15·37-s − 2.34·41-s + 0.745·45-s + 11/7·49-s − 3.29·53-s + 0.372·65-s + 2.10·73-s + 16/9·81-s − 1.19·85-s + 1.48·89-s + 0.203·97-s − 0.696·101-s + 0.383·109-s + 0.0940·113-s + 1.38·117-s + 3/11·121-s − 0.0894·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-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$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.3.a_f
7$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.7.a_al
11$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.11.a_ad
17$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.17.al_ck
19$C_2^2$ \( 1 + 31 T^{2} + p^{2} T^{4} \) 2.19.a_bf
23$C_2^2$ \( 1 + 12 T^{2} + p^{2} T^{4} \) 2.23.a_m
29$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.29.af_w
31$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.31.a_c
37$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.37.h_dg
41$C_2$$\times$$C_2$ \( ( 1 + 5 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.41.p_fc
43$C_2^2$ \( 1 - 11 T^{2} + p^{2} T^{4} \) 2.43.a_al
47$C_2^2$ \( 1 - 65 T^{2} + p^{2} T^{4} \) 2.47.a_acn
53$C_2$$\times$$C_2$ \( ( 1 + 11 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.53.y_jp
59$C_2^2$ \( 1 - 61 T^{2} + p^{2} T^{4} \) 2.59.a_acj
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.61.a_eo
67$C_2^2$ \( 1 - 90 T^{2} + p^{2} T^{4} \) 2.67.a_adm
71$C_2^2$ \( 1 + 140 T^{2} + p^{2} T^{4} \) 2.71.a_fk
73$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) 2.73.as_is
79$C_2^2$ \( 1 - 9 T^{2} + p^{2} T^{4} \) 2.79.a_aj
83$C_2^2$ \( 1 + 64 T^{2} + p^{2} T^{4} \) 2.83.a_cm
89$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + p T^{2} ) \) 2.89.ao_gw
97$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.97.ac_hj
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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

−8.353739195471140903662912812847, −7.965759397747321553569728354974, −7.61902575877467189885178750716, −7.12082937606241218502875229574, −6.42358731620841217089089316493, −6.12751093570791002493963028879, −5.37291033099871729749479548359, −5.16066528268197836608308816682, −4.78270554126984903682747264597, −3.69573193265299988561883081822, −3.33757601779109961661539604746, −2.99139852326999668167494126776, −2.18637396758026576484810367964, −1.16112454676762676307468927544, 0, 1.16112454676762676307468927544, 2.18637396758026576484810367964, 2.99139852326999668167494126776, 3.33757601779109961661539604746, 3.69573193265299988561883081822, 4.78270554126984903682747264597, 5.16066528268197836608308816682, 5.37291033099871729749479548359, 6.12751093570791002493963028879, 6.42358731620841217089089316493, 7.12082937606241218502875229574, 7.61902575877467189885178750716, 7.965759397747321553569728354974, 8.353739195471140903662912812847

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