Properties

Label 2-644-644.251-c0-0-0
Degree $2$
Conductor $644$
Sign $0.702 - 0.711i$
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  + (−0.142 + 0.989i)2-s + (−0.959 − 0.281i)4-s + (0.654 − 0.755i)7-s + (0.415 − 0.909i)8-s + (0.959 − 0.281i)9-s + (−0.239 + 0.153i)11-s + (0.654 + 0.755i)14-s + (0.841 + 0.540i)16-s + (0.142 + 0.989i)18-s + (−0.118 − 0.258i)22-s + (0.959 + 0.281i)23-s + (−0.841 − 0.540i)25-s + (−0.841 + 0.540i)28-s + (−0.698 + 1.53i)29-s + (−0.654 + 0.755i)32-s + ⋯
L(s)  = 1  + (−0.142 + 0.989i)2-s + (−0.959 − 0.281i)4-s + (0.654 − 0.755i)7-s + (0.415 − 0.909i)8-s + (0.959 − 0.281i)9-s + (−0.239 + 0.153i)11-s + (0.654 + 0.755i)14-s + (0.841 + 0.540i)16-s + (0.142 + 0.989i)18-s + (−0.118 − 0.258i)22-s + (0.959 + 0.281i)23-s + (−0.841 − 0.540i)25-s + (−0.841 + 0.540i)28-s + (−0.698 + 1.53i)29-s + (−0.654 + 0.755i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.702 - 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.702 - 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(644\)    =    \(2^{2} \cdot 7 \cdot 23\)
Sign: $0.702 - 0.711i$
Analytic conductor: \(0.321397\)
Root analytic conductor: \(0.566919\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{644} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 644,\ (\ :0),\ 0.702 - 0.711i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8826949935\)
\(L(\frac12)\) \(\approx\) \(0.8826949935\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.142 - 0.989i)T \)
7 \( 1 + (-0.654 + 0.755i)T \)
23 \( 1 + (-0.959 - 0.281i)T \)
good3 \( 1 + (-0.959 + 0.281i)T^{2} \)
5 \( 1 + (0.841 + 0.540i)T^{2} \)
11 \( 1 + (0.239 - 0.153i)T + (0.415 - 0.909i)T^{2} \)
13 \( 1 + (0.142 - 0.989i)T^{2} \)
17 \( 1 + (-0.654 - 0.755i)T^{2} \)
19 \( 1 + (0.654 - 0.755i)T^{2} \)
29 \( 1 + (0.698 - 1.53i)T + (-0.654 - 0.755i)T^{2} \)
31 \( 1 + (-0.959 - 0.281i)T^{2} \)
37 \( 1 + (-0.425 - 1.45i)T + (-0.841 + 0.540i)T^{2} \)
41 \( 1 + (-0.841 - 0.540i)T^{2} \)
43 \( 1 + (0.273 + 1.89i)T + (-0.959 + 0.281i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (0.817 + 0.708i)T + (0.142 + 0.989i)T^{2} \)
59 \( 1 + (-0.142 + 0.989i)T^{2} \)
61 \( 1 + (-0.959 - 0.281i)T^{2} \)
67 \( 1 + (-0.698 - 0.449i)T + (0.415 + 0.909i)T^{2} \)
71 \( 1 + (1.07 - 1.66i)T + (-0.415 - 0.909i)T^{2} \)
73 \( 1 + (0.654 - 0.755i)T^{2} \)
79 \( 1 + (1.10 + 1.27i)T + (-0.142 + 0.989i)T^{2} \)
83 \( 1 + (-0.841 + 0.540i)T^{2} \)
89 \( 1 + (-0.959 + 0.281i)T^{2} \)
97 \( 1 + (0.841 + 0.540i)T^{2} \)
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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

−10.55716886661694276710542243042, −9.988054022570110434596938628236, −9.021501675901416920960187517231, −8.094740585933814248966637715608, −7.25710303709181885466813062710, −6.73967637196956060179019928601, −5.40530295035803749666689886059, −4.58900708027981681281685581273, −3.65497160539527419559723338526, −1.41166859007223781262143576848, 1.61004919740896207520322029994, 2.68099531168247666146408690782, 4.06701421015025900571641161458, 4.91725283921593521517753884374, 5.92088818112540548120614233413, 7.51752071160415773903721518619, 8.118127695084178911188175769709, 9.230945371645260595595232959276, 9.746550694635994754621987598711, 10.88740930608063170584363554009

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