Properties

Label 4-112e2-1.1-c1e2-0-1
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  − 3-s − 5-s − 9-s + 5·11-s + 4·13-s + 15-s − 17-s + 19-s + 3·23-s + 25-s + 7·31-s − 5·33-s + 9·37-s − 4·39-s − 4·41-s − 8·43-s + 45-s − 7·47-s − 7·49-s + 51-s + 5·53-s − 5·55-s − 57-s + 7·59-s + 3·61-s − 4·65-s + 67-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s − 1/3·9-s + 1.50·11-s + 1.10·13-s + 0.258·15-s − 0.242·17-s + 0.229·19-s + 0.625·23-s + 1/5·25-s + 1.25·31-s − 0.870·33-s + 1.47·37-s − 0.640·39-s − 0.624·41-s − 1.21·43-s + 0.149·45-s − 1.02·47-s − 49-s + 0.140·51-s + 0.686·53-s − 0.674·55-s − 0.132·57-s + 0.911·59-s + 0.384·61-s − 0.496·65-s + 0.122·67-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

\(L(1)\) \(\approx\) \(0.8721820425\)
\(L(\frac12)\) \(\approx\) \(0.8721820425\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T^{2} \)
good3$D_{4}$ \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \)
5$D_{4}$ \( 1 + T + p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - 5 T + 18 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
13$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$D_{4}$ \( 1 + T + 12 T^{2} + p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - T - 6 T^{2} - p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 3 T + 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 7 T + 38 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 9 T + 60 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
41$C_4$ \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
47$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$D_{4}$ \( 1 - 5 T + 20 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 7 T + 10 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$D_{4}$ \( 1 - T - 54 T^{2} - p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 8 T + 46 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 3 T + 12 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 3 T + 54 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
89$D_{4}$ \( 1 + 13 T + 124 T^{2} + 13 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
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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

−16.3254484185, −15.9780323483, −15.2824552678, −14.8709787754, −14.4877760383, −13.7864925901, −13.3819142815, −12.8816956304, −12.1501200022, −11.6908461632, −11.2598471936, −11.1685835596, −10.1701213491, −9.70035925712, −9.05119599139, −8.39992155667, −8.13581021341, −7.03515547792, −6.61298948326, −6.10230454409, −5.34727264542, −4.47984542616, −3.83967735094, −2.98334203932, −1.31216934475, 1.31216934475, 2.98334203932, 3.83967735094, 4.47984542616, 5.34727264542, 6.10230454409, 6.61298948326, 7.03515547792, 8.13581021341, 8.39992155667, 9.05119599139, 9.70035925712, 10.1701213491, 11.1685835596, 11.2598471936, 11.6908461632, 12.1501200022, 12.8816956304, 13.3819142815, 13.7864925901, 14.4877760383, 14.8709787754, 15.2824552678, 15.9780323483, 16.3254484185

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