Properties

Label 4-453152-1.1-c1e2-0-3
Degree $4$
Conductor $453152$
Sign $1$
Analytic cond. $28.8933$
Root an. cond. $2.31845$
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  + 2-s + 4-s + 8-s + 2·9-s + 8·13-s + 16-s + 6·17-s + 2·18-s − 10·25-s + 8·26-s + 32-s + 6·34-s + 2·36-s − 49-s − 10·50-s + 8·52-s + 12·53-s + 64-s + 6·68-s + 2·72-s − 5·81-s + 12·89-s − 98-s − 10·100-s + 8·104-s + 12·106-s + 16·117-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.353·8-s + 2/3·9-s + 2.21·13-s + 1/4·16-s + 1.45·17-s + 0.471·18-s − 2·25-s + 1.56·26-s + 0.176·32-s + 1.02·34-s + 1/3·36-s − 1/7·49-s − 1.41·50-s + 1.10·52-s + 1.64·53-s + 1/8·64-s + 0.727·68-s + 0.235·72-s − 5/9·81-s + 1.27·89-s − 0.101·98-s − 100-s + 0.784·104-s + 1.16·106-s + 1.47·117-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 453152 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 453152 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(453152\)    =    \(2^{5} \cdot 7^{2} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(28.8933\)
Root analytic conductor: \(2.31845\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{453152} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 453152,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.821757156\)
\(L(\frac12)\) \(\approx\) \(3.821757156\)
\(L(\frac{3}{2})\) 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)$
bad2$C_1$ \( 1 - T \)
7$C_2$ \( 1 + T^{2} \)
17$C_2$ \( 1 - 6 T + p T^{2} \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
5$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2$ \( ( 1 - p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 - p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
79$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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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

−8.655064726124824394184183303484, −7.975072260169784816600891236211, −7.63282182808416058464170813976, −7.27776442666612344143570168035, −6.53665060632888009337431021869, −6.19002274250768406878646019466, −5.75674549062065341809885026160, −5.42717660586362568204157976090, −4.72476803437976548009006900145, −4.00437577532359061015721712376, −3.73940051357221662734724785910, −3.38179088440528658396943141493, −2.46424226435271921075104708179, −1.64893568141364615823925507110, −1.06693560515841577648054251790, 1.06693560515841577648054251790, 1.64893568141364615823925507110, 2.46424226435271921075104708179, 3.38179088440528658396943141493, 3.73940051357221662734724785910, 4.00437577532359061015721712376, 4.72476803437976548009006900145, 5.42717660586362568204157976090, 5.75674549062065341809885026160, 6.19002274250768406878646019466, 6.53665060632888009337431021869, 7.27776442666612344143570168035, 7.63282182808416058464170813976, 7.975072260169784816600891236211, 8.655064726124824394184183303484

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