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Properties

Label 2-644-644.559-c0-0-0
Degree $2$
Conductor $644$
Sign $0.919 - 0.392i$
Analytic cond. $0.321397$
Root an. cond. $0.566919$
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 + (−0.959 + 0.281i)7-s + (0.841 − 0.540i)8-s + (−0.415 + 0.909i)9-s + (1.10 + 1.27i)11-s + (0.841 + 0.540i)14-s + (−0.959 − 0.281i)16-s + (0.959 − 0.281i)18-s + (0.239 − 1.66i)22-s + (0.415 + 0.909i)23-s + (0.654 − 0.755i)25-s + (−0.142 − 0.989i)28-s + (−0.186 + 1.29i)29-s + (0.415 + 0.909i)32-s + ⋯
L(s)  = 1  + (−0.654 − 0.755i)2-s + (−0.142 + 0.989i)4-s + (−0.959 + 0.281i)7-s + (0.841 − 0.540i)8-s + (−0.415 + 0.909i)9-s + (1.10 + 1.27i)11-s + (0.841 + 0.540i)14-s + (−0.959 − 0.281i)16-s + (0.959 − 0.281i)18-s + (0.239 − 1.66i)22-s + (0.415 + 0.909i)23-s + (0.654 − 0.755i)25-s + (−0.142 − 0.989i)28-s + (−0.186 + 1.29i)29-s + (0.415 + 0.909i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.919 - 0.392i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.919 - 0.392i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(644\)    =    \(2^{2} \cdot 7 \cdot 23\)
Sign: $0.919 - 0.392i$
Analytic conductor: \(0.321397\)
Root analytic conductor: \(0.566919\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{644} (559, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 644,\ (\ :0),\ 0.919 - 0.392i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5900897747\)
\(L(\frac12)\) \(\approx\) \(0.5900897747\)
\(L(1)\) 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 \( 1 + (0.959 - 0.281i)T \)
23 \( 1 + (-0.415 - 0.909i)T \)
good3 \( 1 + (0.415 - 0.909i)T^{2} \)
5 \( 1 + (-0.654 + 0.755i)T^{2} \)
11 \( 1 + (-1.10 - 1.27i)T + (-0.142 + 0.989i)T^{2} \)
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 \( 1 + (0.186 - 1.29i)T + (-0.959 - 0.281i)T^{2} \)
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 \( 1 + (-0.698 + 0.449i)T + (0.415 - 0.909i)T^{2} \)
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 \( 1 + (0.186 - 0.215i)T + (-0.142 - 0.989i)T^{2} \)
71 \( 1 + (-0.817 - 0.708i)T + (0.142 + 0.989i)T^{2} \)
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

−10.72903260424697544079592693869, −9.924184084142438597935648003081, −9.235108423954795057282310129639, −8.522171466762660008881792341698, −7.32776460808480176744802113936, −6.70950811635892691012171654147, −5.21539064037664156272850239579, −4.04483504661900090591572218572, −2.93815142393270950057337894038, −1.77297548407386689065757158046, 0.885394458758629904411216923233, 3.06429546428394004828212590839, 4.19119309908013955105656220071, 5.75420664283948622794241551416, 6.34679860751745611316295436980, 7.00202267017010001489028306214, 8.241356294627002491731473525070, 9.099970921798650655370846412729, 9.457542998260953937464379781058, 10.59960150283930283244610492971

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