Properties

Label 4-1088-1.1-c1e2-0-0
Degree $4$
Conductor $1088$
Sign $1$
Analytic cond. $0.0693718$
Root an. cond. $0.513210$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

This L-function arises from both an elliptic curve over a number field, and from a genus 2 curve over the rationals. It is probably the L-function of smallest conductor with that property.

Dirichlet series

L(s)  = 1  − 2·4-s − 2·7-s − 2·9-s + 4·16-s + 5·17-s − 6·23-s + 2·25-s + 4·28-s − 2·31-s + 4·36-s + 12·47-s − 2·49-s + 4·63-s − 8·64-s − 10·68-s + 6·71-s − 20·73-s − 2·79-s − 5·81-s + 12·92-s − 8·97-s − 4·100-s − 8·103-s − 8·112-s + 24·113-s − 10·119-s − 10·121-s + ⋯
L(s)  = 1  − 4-s − 0.755·7-s − 2/3·9-s + 16-s + 1.21·17-s − 1.25·23-s + 2/5·25-s + 0.755·28-s − 0.359·31-s + 2/3·36-s + 1.75·47-s − 2/7·49-s + 0.503·63-s − 64-s − 1.21·68-s + 0.712·71-s − 2.34·73-s − 0.225·79-s − 5/9·81-s + 1.25·92-s − 0.812·97-s − 2/5·100-s − 0.788·103-s − 0.755·112-s + 2.25·113-s − 0.916·119-s − 0.909·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1088\)    =    \(2^{6} \cdot 17\)
Sign: $1$
Analytic conductor: \(0.0693718\)
Root analytic conductor: \(0.513210\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1088} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1088,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4366701692\)
\(L(\frac12)\) \(\approx\) \(0.4366701692\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T^{2} \)
17$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 6 T + p T^{2} ) \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
7$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
71$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
73$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
79$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
83$C_2^2$ \( 1 - 86 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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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

−14.19044829739582126239819958846, −13.54189737490713903220419819081, −12.86589430836834663288318270965, −12.29726535287971277212019804411, −11.76221275592136857524468249377, −10.71210139881204999206150662240, −10.05681574304479561828369141584, −9.514581676057178887948389923580, −8.766450619065699860669585506452, −8.104427830147069626222932574670, −7.25728763856226074637060722968, −6.01912712781998810706342352491, −5.48237499609119068897650555264, −4.21524744281209647906717432747, −3.17400203367696839713552335058, 3.17400203367696839713552335058, 4.21524744281209647906717432747, 5.48237499609119068897650555264, 6.01912712781998810706342352491, 7.25728763856226074637060722968, 8.104427830147069626222932574670, 8.766450619065699860669585506452, 9.514581676057178887948389923580, 10.05681574304479561828369141584, 10.71210139881204999206150662240, 11.76221275592136857524468249377, 12.29726535287971277212019804411, 12.86589430836834663288318270965, 13.54189737490713903220419819081, 14.19044829739582126239819958846

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