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Properties

Label 2-644-644.447-c0-0-1
Degree $2$
Conductor $644$
Sign $0.561 + 0.827i$
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  + (0.841 − 0.540i)2-s + (0.415 − 0.909i)4-s + (0.959 + 0.281i)7-s + (−0.142 − 0.989i)8-s + (−0.415 − 0.909i)9-s + (−1.10 + 1.27i)11-s + (0.959 − 0.281i)14-s + (−0.654 − 0.755i)16-s + (−0.841 − 0.540i)18-s + (−0.239 + 1.66i)22-s + (−0.415 + 0.909i)23-s + (0.654 + 0.755i)25-s + (0.654 − 0.755i)28-s + (−0.186 − 1.29i)29-s + (−0.959 − 0.281i)32-s + ⋯
L(s)  = 1  + (0.841 − 0.540i)2-s + (0.415 − 0.909i)4-s + (0.959 + 0.281i)7-s + (−0.142 − 0.989i)8-s + (−0.415 − 0.909i)9-s + (−1.10 + 1.27i)11-s + (0.959 − 0.281i)14-s + (−0.654 − 0.755i)16-s + (−0.841 − 0.540i)18-s + (−0.239 + 1.66i)22-s + (−0.415 + 0.909i)23-s + (0.654 + 0.755i)25-s + (0.654 − 0.755i)28-s + (−0.186 − 1.29i)29-s + (−0.959 − 0.281i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.561 + 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.561 + 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(644\)    =    \(2^{2} \cdot 7 \cdot 23\)
Sign: $0.561 + 0.827i$
Analytic conductor: \(0.321397\)
Root analytic conductor: \(0.566919\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{644} (447, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 644,\ (\ :0),\ 0.561 + 0.827i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.458437956\)
\(L(\frac12)\) \(\approx\) \(1.458437956\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.841 + 0.540i)T \)
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

−10.80297239495076326478600151587, −9.945290682249983852214441174158, −9.146078762126913999971393391962, −7.898608280815964919365260960851, −7.01235408505281861519190305974, −5.80067474358589026849076105448, −5.08095132919396014268820296540, −4.14109622249776931501680517051, −2.86879480220487331381541703599, −1.74210575059876574475919925533, 2.27365781376578723709542965353, 3.37009940362141078419149190936, 4.78521077952654514638564931924, 5.25136188891817880105833340253, 6.28755693346023740976045197636, 7.43682197475437509583956451284, 8.265400999236635506000233410465, 8.583732084540483701183069496038, 10.53145741349455491349284387267, 10.90302624002424559720903340406

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