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Properties

Label 4-226e2-1.1-c1e2-0-3
Degree $4$
Conductor $51076$
Sign $1$
Analytic cond. $3.25665$
Root an. cond. $1.34336$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3·4-s + 4·8-s + 6·9-s − 8·11-s + 4·13-s + 5·16-s + 12·18-s − 16·22-s + 2·25-s + 8·26-s − 16·31-s + 6·32-s + 18·36-s − 4·41-s − 24·44-s − 14·49-s + 4·50-s + 12·52-s − 12·53-s + 28·61-s − 32·62-s + 7·64-s + 24·72-s + 27·81-s − 8·82-s + 8·83-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s + 1.41·8-s + 2·9-s − 2.41·11-s + 1.10·13-s + 5/4·16-s + 2.82·18-s − 3.41·22-s + 2/5·25-s + 1.56·26-s − 2.87·31-s + 1.06·32-s + 3·36-s − 0.624·41-s − 3.61·44-s − 2·49-s + 0.565·50-s + 1.66·52-s − 1.64·53-s + 3.58·61-s − 4.06·62-s + 7/8·64-s + 2.82·72-s + 3·81-s − 0.883·82-s + 0.878·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 51076 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51076 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(51076\)    =    \(2^{2} \cdot 113^{2}\)
Sign: $1$
Analytic conductor: \(3.25665\)
Root analytic conductor: \(1.34336\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 51076,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.352874252\)
\(L(\frac12)\) \(\approx\) \(3.352874252\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
113$C_2$ \( 1 - 18 T + p T^{2} \)
good3$C_2$ \( ( 1 - p T^{2} )^{2} \)
5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 66 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 54 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 86 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 134 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 42 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
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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

−12.56148106321288342115959653321, −12.52705491805036900993902506504, −11.30496556292223352835783277564, −11.25927233544557362048783816546, −10.56738073849716309346230716057, −10.35888933863551670885832556900, −9.789874225791298249819900340442, −9.186129281741545725906331388646, −8.129027596445550267391889632932, −8.009190898945144347091161058434, −7.13315295681790457124626290988, −7.07832910723804054651330231723, −6.24062611508485339120063978653, −5.60260353828375197569700597266, −4.92470629433185650636121682668, −4.83288758172427023096362729954, −3.64758202452903894921898714281, −3.57306096681819378549829252385, −2.37644293769007538030921825811, −1.64081282802269209041032077337, 1.64081282802269209041032077337, 2.37644293769007538030921825811, 3.57306096681819378549829252385, 3.64758202452903894921898714281, 4.83288758172427023096362729954, 4.92470629433185650636121682668, 5.60260353828375197569700597266, 6.24062611508485339120063978653, 7.07832910723804054651330231723, 7.13315295681790457124626290988, 8.009190898945144347091161058434, 8.129027596445550267391889632932, 9.186129281741545725906331388646, 9.789874225791298249819900340442, 10.35888933863551670885832556900, 10.56738073849716309346230716057, 11.25927233544557362048783816546, 11.30496556292223352835783277564, 12.52705491805036900993902506504, 12.56148106321288342115959653321

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