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Properties

Label 2-1288-1288.419-c0-0-0
Degree $2$
Conductor $1288$
Sign $-0.298 - 0.954i$
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  + (0.654 + 0.755i)2-s + (−0.142 + 0.989i)4-s + (0.415 + 0.909i)7-s + (−0.841 + 0.540i)8-s + (−0.142 − 0.989i)9-s + (0.425 + 1.45i)11-s + (−0.415 + 0.909i)14-s + (−0.959 − 0.281i)16-s + (0.654 − 0.755i)18-s + (−0.817 + 1.27i)22-s + (0.142 − 0.989i)23-s + (−0.959 − 0.281i)25-s + (−0.959 + 0.281i)28-s + (0.304 + 0.474i)29-s + (−0.415 − 0.909i)32-s + ⋯
L(s)  = 1  + (0.654 + 0.755i)2-s + (−0.142 + 0.989i)4-s + (0.415 + 0.909i)7-s + (−0.841 + 0.540i)8-s + (−0.142 − 0.989i)9-s + (0.425 + 1.45i)11-s + (−0.415 + 0.909i)14-s + (−0.959 − 0.281i)16-s + (0.654 − 0.755i)18-s + (−0.817 + 1.27i)22-s + (0.142 − 0.989i)23-s + (−0.959 − 0.281i)25-s + (−0.959 + 0.281i)28-s + (0.304 + 0.474i)29-s + (−0.415 − 0.909i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.298 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.298 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1288\)    =    \(2^{3} \cdot 7 \cdot 23\)
Sign: $-0.298 - 0.954i$
Analytic conductor: \(0.642795\)
Root analytic conductor: \(0.801745\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1288} (419, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1288,\ (\ :0),\ -0.298 - 0.954i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.470932840\)
\(L(\frac12)\) \(\approx\) \(1.470932840\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.654 - 0.755i)T \)
7 \( 1 + (-0.415 - 0.909i)T \)
23 \( 1 + (-0.142 + 0.989i)T \)
good3 \( 1 + (0.142 + 0.989i)T^{2} \)
5 \( 1 + (0.959 + 0.281i)T^{2} \)
11 \( 1 + (-0.425 - 1.45i)T + (-0.841 + 0.540i)T^{2} \)
13 \( 1 + (-0.654 - 0.755i)T^{2} \)
17 \( 1 + (0.415 - 0.909i)T^{2} \)
19 \( 1 + (0.415 + 0.909i)T^{2} \)
29 \( 1 + (-0.304 - 0.474i)T + (-0.415 + 0.909i)T^{2} \)
31 \( 1 + (-0.142 + 0.989i)T^{2} \)
37 \( 1 + (-0.118 - 0.822i)T + (-0.959 + 0.281i)T^{2} \)
41 \( 1 + (0.959 + 0.281i)T^{2} \)
43 \( 1 + (-1.49 - 1.29i)T + (0.142 + 0.989i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (0.797 + 1.74i)T + (-0.654 + 0.755i)T^{2} \)
59 \( 1 + (0.654 + 0.755i)T^{2} \)
61 \( 1 + (0.142 - 0.989i)T^{2} \)
67 \( 1 + (-0.304 + 1.03i)T + (-0.841 - 0.540i)T^{2} \)
71 \( 1 + (-0.425 + 1.45i)T + (-0.841 - 0.540i)T^{2} \)
73 \( 1 + (-0.415 - 0.909i)T^{2} \)
79 \( 1 + (-0.797 + 1.74i)T + (-0.654 - 0.755i)T^{2} \)
83 \( 1 + (-0.959 + 0.281i)T^{2} \)
89 \( 1 + (-0.142 - 0.989i)T^{2} \)
97 \( 1 + (-0.959 - 0.281i)T^{2} \)
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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

−9.756605906440448950140794134425, −9.204051638983277372059902837963, −8.364820917350416483277886243487, −7.58280860214368738197618040254, −6.57058429808750276313191617402, −6.12704832436696322858354570724, −4.96684782853166550899409214781, −4.35971967752148494471008823287, −3.21304619955918227199451312172, −2.06983046213629488884806865892, 1.13025111430425478292664008984, 2.40247000889854054862594900355, 3.60287668309645526978096877270, 4.24969573102440266426131001497, 5.38820166178428089841677816727, 5.93422413105988092500122348966, 7.15737536168417883750522664376, 8.003301960966503408123792426492, 8.940545616300611211280732871418, 9.836999497321101657817677668814

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