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Properties

Label 2-1288-1288.1189-c0-0-2
Degree $2$
Conductor $1288$
Sign $0.904 + 0.426i$
Analytic cond. $0.642795$
Root an. cond. $0.801745$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.841 − 0.540i)2-s + (0.415 + 0.909i)4-s + (0.959 − 0.281i)7-s + (0.142 − 0.989i)8-s + (−0.415 + 0.909i)9-s + (0.817 − 0.708i)11-s + (−0.959 − 0.281i)14-s + (−0.654 + 0.755i)16-s + (0.841 − 0.540i)18-s + (−1.07 + 0.153i)22-s + (0.415 + 0.909i)23-s + (0.654 − 0.755i)25-s + (0.654 + 0.755i)28-s + (−1.49 − 0.215i)29-s + (0.959 − 0.281i)32-s + ⋯
L(s)  = 1  + (−0.841 − 0.540i)2-s + (0.415 + 0.909i)4-s + (0.959 − 0.281i)7-s + (0.142 − 0.989i)8-s + (−0.415 + 0.909i)9-s + (0.817 − 0.708i)11-s + (−0.959 − 0.281i)14-s + (−0.654 + 0.755i)16-s + (0.841 − 0.540i)18-s + (−1.07 + 0.153i)22-s + (0.415 + 0.909i)23-s + (0.654 − 0.755i)25-s + (0.654 + 0.755i)28-s + (−1.49 − 0.215i)29-s + (0.959 − 0.281i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 + 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 + 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1288\)    =    \(2^{3} \cdot 7 \cdot 23\)
Sign: $0.904 + 0.426i$
Analytic conductor: \(0.642795\)
Root analytic conductor: \(0.801745\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1288} (1189, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1288,\ (\ :0),\ 0.904 + 0.426i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8244299603\)
\(L(\frac12)\) \(\approx\) \(0.8244299603\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.841 + 0.540i)T \)
7 \( 1 + (-0.959 + 0.281i)T \)
23 \( 1 + (-0.415 - 0.909i)T \)
good3 \( 1 + (0.415 - 0.909i)T^{2} \)
5 \( 1 + (-0.654 + 0.755i)T^{2} \)
11 \( 1 + (-0.817 + 0.708i)T + (0.142 - 0.989i)T^{2} \)
13 \( 1 + (0.841 + 0.540i)T^{2} \)
17 \( 1 + (0.959 + 0.281i)T^{2} \)
19 \( 1 + (-0.959 + 0.281i)T^{2} \)
29 \( 1 + (1.49 + 0.215i)T + (0.959 + 0.281i)T^{2} \)
31 \( 1 + (-0.415 - 0.909i)T^{2} \)
37 \( 1 + (-0.512 - 0.234i)T + (0.654 + 0.755i)T^{2} \)
41 \( 1 + (0.654 - 0.755i)T^{2} \)
43 \( 1 + (-0.983 - 1.53i)T + (-0.415 + 0.909i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (0.425 + 1.45i)T + (-0.841 + 0.540i)T^{2} \)
59 \( 1 + (0.841 + 0.540i)T^{2} \)
61 \( 1 + (0.415 + 0.909i)T^{2} \)
67 \( 1 + (-1.49 - 1.29i)T + (0.142 + 0.989i)T^{2} \)
71 \( 1 + (-1.10 + 1.27i)T + (-0.142 - 0.989i)T^{2} \)
73 \( 1 + (0.959 - 0.281i)T^{2} \)
79 \( 1 + (1.25 + 0.368i)T + (0.841 + 0.540i)T^{2} \)
83 \( 1 + (-0.654 - 0.755i)T^{2} \)
89 \( 1 + (-0.415 + 0.909i)T^{2} \)
97 \( 1 + (0.654 - 0.755i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.756094220411897049447984727227, −8.991714856625012488857041431135, −8.196505147479242335506646672807, −7.71771774730722092724911694482, −6.75318045348413270673859636453, −5.62847573976686767091997934735, −4.52979268686202283509391104906, −3.52951484404720809550837005974, −2.35416134494427293773469962511, −1.25806081678372257871409609691, 1.23823392435225361029897719663, 2.41511992798911031838341600002, 3.97121273439975976229284932730, 5.08231944681155929186289742051, 5.88185190711044411879090857892, 6.82476079979522188197818680071, 7.43974032973640332974071274799, 8.438243951336982497382739090420, 9.107198945050977500669679480615, 9.530175871973256455612044991043

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