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Properties

Label 2-1564-1564.611-c0-0-3
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: $\chi_{1564} (611, \cdot )$
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

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

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