Properties

Label 4-112e2-1.1-c1e2-0-5
Degree $4$
Conductor $12544$
Sign $1$
Analytic cond. $0.799816$
Root an. cond. $0.945687$
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  + 4·7-s + 2·9-s − 4·11-s − 8·23-s + 2·25-s + 4·29-s + 4·37-s − 4·43-s + 9·49-s − 12·53-s + 8·63-s − 12·67-s + 8·71-s − 16·77-s + 16·79-s − 5·81-s − 8·99-s − 4·107-s + 4·109-s − 12·113-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯
L(s)  = 1  + 1.51·7-s + 2/3·9-s − 1.20·11-s − 1.66·23-s + 2/5·25-s + 0.742·29-s + 0.657·37-s − 0.609·43-s + 9/7·49-s − 1.64·53-s + 1.00·63-s − 1.46·67-s + 0.949·71-s − 1.82·77-s + 1.80·79-s − 5/9·81-s − 0.804·99-s − 0.386·107-s + 0.383·109-s − 1.12·113-s − 0.545·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12544 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(12544\)    =    \(2^{8} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(0.799816\)
Root analytic conductor: \(0.945687\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{12544} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 12544,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.164597616\)
\(L(\frac12)\) \(\approx\) \(1.164597616\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_2$ \( 1 - 4 T + p T^{2} \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
11$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
59$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
73$C_2^2$ \( 1 + 94 T^{2} + p^{2} T^{4} \)
79$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
83$C_2^2$ \( 1 + 94 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 66 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
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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

−11.24546137874711584465944614206, −10.74722931410243329022959016146, −10.27742252335714100572729688138, −9.798954659172424338252434247629, −9.033478917686801892935856391267, −8.246492846934531661690789080203, −7.933457477097933243038357226368, −7.55981727192192512249238094619, −6.65030690960747568704415684834, −5.94330478058426307240944575384, −5.07779563458741115331533479450, −4.71274112300962733613180106864, −3.90687865497311603496630111094, −2.65445017208115652411046904447, −1.66003289639842380345136186146, 1.66003289639842380345136186146, 2.65445017208115652411046904447, 3.90687865497311603496630111094, 4.71274112300962733613180106864, 5.07779563458741115331533479450, 5.94330478058426307240944575384, 6.65030690960747568704415684834, 7.55981727192192512249238094619, 7.933457477097933243038357226368, 8.246492846934531661690789080203, 9.033478917686801892935856391267, 9.798954659172424338252434247629, 10.27742252335714100572729688138, 10.74722931410243329022959016146, 11.24546137874711584465944614206

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