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Properties

Label 2-644-644.83-c0-0-1
Degree $2$
Conductor $644$
Sign $0.264 + 0.964i$
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  + (−0.959 − 0.281i)2-s + (0.841 + 0.540i)4-s + (0.415 − 0.909i)7-s + (−0.654 − 0.755i)8-s + (0.142 − 0.989i)9-s + (−1.25 − 0.368i)11-s + (−0.654 + 0.755i)14-s + (0.415 + 0.909i)16-s + (−0.415 + 0.909i)18-s + (1.10 + 0.708i)22-s + (−0.142 − 0.989i)23-s + (0.959 − 0.281i)25-s + (0.841 − 0.540i)28-s + (1.61 + 1.03i)29-s + (−0.142 − 0.989i)32-s + ⋯
L(s)  = 1  + (−0.959 − 0.281i)2-s + (0.841 + 0.540i)4-s + (0.415 − 0.909i)7-s + (−0.654 − 0.755i)8-s + (0.142 − 0.989i)9-s + (−1.25 − 0.368i)11-s + (−0.654 + 0.755i)14-s + (0.415 + 0.909i)16-s + (−0.415 + 0.909i)18-s + (1.10 + 0.708i)22-s + (−0.142 − 0.989i)23-s + (0.959 − 0.281i)25-s + (0.841 − 0.540i)28-s + (1.61 + 1.03i)29-s + (−0.142 − 0.989i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.264 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.264 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(644\)    =    \(2^{2} \cdot 7 \cdot 23\)
Sign: $0.264 + 0.964i$
Analytic conductor: \(0.321397\)
Root analytic conductor: \(0.566919\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{644} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 644,\ (\ :0),\ 0.264 + 0.964i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5942116449\)
\(L(\frac12)\) \(\approx\) \(0.5942116449\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.959 + 0.281i)T \)
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 \( 1 + (1.25 + 0.368i)T + (0.841 + 0.540i)T^{2} \)
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 \( 1 + (-1.61 - 1.03i)T + (0.415 + 0.909i)T^{2} \)
31 \( 1 + (-0.142 - 0.989i)T^{2} \)
37 \( 1 + (1.80 + 0.258i)T + (0.959 + 0.281i)T^{2} \)
41 \( 1 + (0.959 - 0.281i)T^{2} \)
43 \( 1 + (-0.186 - 0.215i)T + (-0.142 + 0.989i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (-0.512 - 0.234i)T + (0.654 + 0.755i)T^{2} \)
59 \( 1 + (-0.654 + 0.755i)T^{2} \)
61 \( 1 + (-0.142 - 0.989i)T^{2} \)
67 \( 1 + (-1.61 + 0.474i)T + (0.841 - 0.540i)T^{2} \)
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

−10.52696370836992454374414825984, −9.932354118120941966285645573318, −8.707709211742382346927890115070, −8.230920724596450959705333601923, −7.11039432300326597836829137168, −6.54168099355989227207048677182, −5.06748611798613211944697357102, −3.73961974253959448435030345153, −2.65817140571111455264899115210, −0.952837224183235232463082720439, 1.89364030733974007384833458832, 2.81111660532146483926635998191, 4.95238390413649211298458676733, 5.46716175919257321744828102898, 6.71276129354884281382858015689, 7.75064148593941228571584239836, 8.231461328071374451410396312424, 9.119497221718384241609565698306, 10.17332689025626757073298149365, 10.65829103679777185514566510482

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