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Properties

Label 2-644-644.475-c0-0-1
Degree $2$
Conductor $644$
Sign $-0.552 + 0.833i$
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  + (0.415 − 0.909i)2-s + (−0.654 − 0.755i)4-s + (−0.841 − 0.540i)7-s + (−0.959 + 0.281i)8-s + (0.654 − 0.755i)9-s + (−0.118 − 0.822i)11-s + (−0.841 + 0.540i)14-s + (−0.142 + 0.989i)16-s + (−0.415 − 0.909i)18-s + (−0.797 − 0.234i)22-s + (0.654 + 0.755i)23-s + (0.142 − 0.989i)25-s + (0.142 + 0.989i)28-s + (−0.273 + 0.0801i)29-s + (0.841 + 0.540i)32-s + ⋯
L(s)  = 1  + (0.415 − 0.909i)2-s + (−0.654 − 0.755i)4-s + (−0.841 − 0.540i)7-s + (−0.959 + 0.281i)8-s + (0.654 − 0.755i)9-s + (−0.118 − 0.822i)11-s + (−0.841 + 0.540i)14-s + (−0.142 + 0.989i)16-s + (−0.415 − 0.909i)18-s + (−0.797 − 0.234i)22-s + (0.654 + 0.755i)23-s + (0.142 − 0.989i)25-s + (0.142 + 0.989i)28-s + (−0.273 + 0.0801i)29-s + (0.841 + 0.540i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.552 + 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.552 + 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(644\)    =    \(2^{2} \cdot 7 \cdot 23\)
Sign: $-0.552 + 0.833i$
Analytic conductor: \(0.321397\)
Root analytic conductor: \(0.566919\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{644} (475, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 644,\ (\ :0),\ -0.552 + 0.833i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9692037207\)
\(L(\frac12)\) \(\approx\) \(0.9692037207\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.415 + 0.909i)T \)
7 \( 1 + (0.841 + 0.540i)T \)
23 \( 1 + (-0.654 - 0.755i)T \)
good3 \( 1 + (-0.654 + 0.755i)T^{2} \)
5 \( 1 + (-0.142 + 0.989i)T^{2} \)
11 \( 1 + (0.118 + 0.822i)T + (-0.959 + 0.281i)T^{2} \)
13 \( 1 + (-0.415 + 0.909i)T^{2} \)
17 \( 1 + (0.841 - 0.540i)T^{2} \)
19 \( 1 + (-0.841 - 0.540i)T^{2} \)
29 \( 1 + (0.273 - 0.0801i)T + (0.841 - 0.540i)T^{2} \)
31 \( 1 + (-0.654 - 0.755i)T^{2} \)
37 \( 1 + (-0.817 - 0.708i)T + (0.142 + 0.989i)T^{2} \)
41 \( 1 + (0.142 - 0.989i)T^{2} \)
43 \( 1 + (-0.544 - 1.19i)T + (-0.654 + 0.755i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (1.07 - 1.66i)T + (-0.415 - 0.909i)T^{2} \)
59 \( 1 + (0.415 - 0.909i)T^{2} \)
61 \( 1 + (-0.654 - 0.755i)T^{2} \)
67 \( 1 + (-0.273 + 1.89i)T + (-0.959 - 0.281i)T^{2} \)
71 \( 1 + (-1.80 - 0.258i)T + (0.959 + 0.281i)T^{2} \)
73 \( 1 + (-0.841 - 0.540i)T^{2} \)
79 \( 1 + (0.239 - 0.153i)T + (0.415 - 0.909i)T^{2} \)
83 \( 1 + (0.142 + 0.989i)T^{2} \)
89 \( 1 + (-0.654 + 0.755i)T^{2} \)
97 \( 1 + (-0.142 + 0.989i)T^{2} \)
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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

−10.58745533598384527220885477609, −9.649651513763944191001627540665, −9.214790036010856890530396751708, −7.936482723947063389330677220462, −6.63882486976685627439062201197, −5.97110486470105728297898706401, −4.64751082407988580599695098127, −3.69370958744439064356073814291, −2.88087373688414571315150999048, −1.05746836958542969239284626486, 2.42310386482465050497776102715, 3.74698686969377715478192995498, 4.82124785268786954012190456950, 5.62177801066749771491521700016, 6.76049505301478037882514081212, 7.31262133197626150505143323109, 8.323330932532848046057096311975, 9.292890337571181266263154367733, 9.933556515823368701424538350188, 11.08385016056744693212739645328

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