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Properties

Label 2-1564-1564.407-c0-0-0
Degree $2$
Conductor $1564$
Sign $0.153 - 0.988i$
Analytic cond. $0.780537$
Root an. cond. $0.883480$
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 + (−0.474 + 0.304i)3-s + (−0.142 − 0.989i)4-s + (−0.0801 + 0.557i)6-s + (−1.89 + 0.557i)7-s + (−0.841 − 0.540i)8-s + (−0.283 + 0.620i)9-s + (0.708 + 0.817i)11-s + (0.368 + 0.425i)12-s + (−0.273 − 0.0801i)13-s + (−0.822 + 1.80i)14-s + (−0.959 + 0.281i)16-s + (−0.142 + 0.989i)17-s + (0.283 + 0.620i)18-s + (0.730 − 0.843i)21-s + 1.08·22-s + ⋯
L(s)  = 1  + (0.654 − 0.755i)2-s + (−0.474 + 0.304i)3-s + (−0.142 − 0.989i)4-s + (−0.0801 + 0.557i)6-s + (−1.89 + 0.557i)7-s + (−0.841 − 0.540i)8-s + (−0.283 + 0.620i)9-s + (0.708 + 0.817i)11-s + (0.368 + 0.425i)12-s + (−0.273 − 0.0801i)13-s + (−0.822 + 1.80i)14-s + (−0.959 + 0.281i)16-s + (−0.142 + 0.989i)17-s + (0.283 + 0.620i)18-s + (0.730 − 0.843i)21-s + 1.08·22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1564 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.153 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1564 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.153 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1564\)    =    \(2^{2} \cdot 17 \cdot 23\)
Sign: $0.153 - 0.988i$
Analytic conductor: \(0.780537\)
Root analytic conductor: \(0.883480\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1564} (407, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1564,\ (\ :0),\ 0.153 - 0.988i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5484797046\)
\(L(\frac12)\) \(\approx\) \(0.5484797046\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.654 + 0.755i)T \)
17 \( 1 + (0.142 - 0.989i)T \)
23 \( 1 + (0.989 + 0.142i)T \)
good3 \( 1 + (0.474 - 0.304i)T + (0.415 - 0.909i)T^{2} \)
5 \( 1 + (0.654 - 0.755i)T^{2} \)
7 \( 1 + (1.89 - 0.557i)T + (0.841 - 0.540i)T^{2} \)
11 \( 1 + (-0.708 - 0.817i)T + (-0.142 + 0.989i)T^{2} \)
13 \( 1 + (0.273 + 0.0801i)T + (0.841 + 0.540i)T^{2} \)
19 \( 1 + (0.959 - 0.281i)T^{2} \)
29 \( 1 + (0.959 + 0.281i)T^{2} \)
31 \( 1 + (-0.909 - 0.584i)T + (0.415 + 0.909i)T^{2} \)
37 \( 1 + (0.654 + 0.755i)T^{2} \)
41 \( 1 + (0.654 - 0.755i)T^{2} \)
43 \( 1 + (-0.415 + 0.909i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (1.61 - 0.474i)T + (0.841 - 0.540i)T^{2} \)
59 \( 1 + (-0.841 - 0.540i)T^{2} \)
61 \( 1 + (-0.415 - 0.909i)T^{2} \)
67 \( 1 + (0.142 + 0.989i)T^{2} \)
71 \( 1 + (-0.142 - 0.989i)T^{2} \)
73 \( 1 + (0.959 - 0.281i)T^{2} \)
79 \( 1 + (1.45 + 0.425i)T + (0.841 + 0.540i)T^{2} \)
83 \( 1 + (0.654 + 0.755i)T^{2} \)
89 \( 1 + (1.41 - 0.909i)T + (0.415 - 0.909i)T^{2} \)
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

−9.895342680300980822562023099889, −9.470218570573009045820106360634, −8.472308806775114472376014136549, −7.07097414926485500741685772897, −6.16365248549437448541632104866, −5.83100214070108160747073729670, −4.69059644151428260391002873013, −3.85532830222358134128339201985, −2.94575696625148721688152904385, −1.91875360758388003423841201394, 0.34654367221916445136474197420, 2.82993950472443707554695996578, 3.55376131097801495017142961731, 4.35441509277942598923315854175, 5.75351752201187914928747902376, 6.25483867840650061503524608967, 6.71676682061838875517811226176, 7.50923505281954307300724734229, 8.596275367147760583486622148731, 9.450834292837391369187200319015

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