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Properties

Label 2-1564-1564.1223-c0-0-3
Degree $2$
Conductor $1564$
Sign $-0.305 + 0.952i$
Analytic cond. $0.780537$
Root an. cond. $0.883480$
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 + (1.89 − 0.557i)3-s + (−0.654 + 0.755i)4-s + (−1.29 − 1.49i)6-s + (−0.215 − 1.49i)7-s + (0.959 + 0.281i)8-s + (2.45 − 1.57i)9-s + (−0.234 + 0.512i)11-s + (−0.822 + 1.80i)12-s + (−0.186 + 1.29i)13-s + (−1.27 + 0.817i)14-s + (−0.142 − 0.989i)16-s + (−0.654 − 0.755i)17-s + (−2.45 − 1.57i)18-s + (−1.24 − 2.72i)21-s + 0.563·22-s + ⋯
L(s)  = 1  + (−0.415 − 0.909i)2-s + (1.89 − 0.557i)3-s + (−0.654 + 0.755i)4-s + (−1.29 − 1.49i)6-s + (−0.215 − 1.49i)7-s + (0.959 + 0.281i)8-s + (2.45 − 1.57i)9-s + (−0.234 + 0.512i)11-s + (−0.822 + 1.80i)12-s + (−0.186 + 1.29i)13-s + (−1.27 + 0.817i)14-s + (−0.142 − 0.989i)16-s + (−0.654 − 0.755i)17-s + (−2.45 − 1.57i)18-s + (−1.24 − 2.72i)21-s + 0.563·22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1564 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.305 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1564 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.305 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1564\)    =    \(2^{2} \cdot 17 \cdot 23\)
Sign: $-0.305 + 0.952i$
Analytic conductor: \(0.780537\)
Root analytic conductor: \(0.883480\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1564} (1223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1564,\ (\ :0),\ -0.305 + 0.952i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.560161940\)
\(L(\frac12)\) \(\approx\) \(1.560161940\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.415 + 0.909i)T \)
17 \( 1 + (0.654 + 0.755i)T \)
23 \( 1 + (0.755 - 0.654i)T \)
good3 \( 1 + (-1.89 + 0.557i)T + (0.841 - 0.540i)T^{2} \)
5 \( 1 + (-0.415 - 0.909i)T^{2} \)
7 \( 1 + (0.215 + 1.49i)T + (-0.959 + 0.281i)T^{2} \)
11 \( 1 + (0.234 - 0.512i)T + (-0.654 - 0.755i)T^{2} \)
13 \( 1 + (0.186 - 1.29i)T + (-0.959 - 0.281i)T^{2} \)
19 \( 1 + (0.142 + 0.989i)T^{2} \)
29 \( 1 + (0.142 - 0.989i)T^{2} \)
31 \( 1 + (0.540 + 0.158i)T + (0.841 + 0.540i)T^{2} \)
37 \( 1 + (-0.415 + 0.909i)T^{2} \)
41 \( 1 + (-0.415 - 0.909i)T^{2} \)
43 \( 1 + (-0.841 + 0.540i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (-0.273 - 1.89i)T + (-0.959 + 0.281i)T^{2} \)
59 \( 1 + (0.959 + 0.281i)T^{2} \)
61 \( 1 + (-0.841 - 0.540i)T^{2} \)
67 \( 1 + (0.654 - 0.755i)T^{2} \)
71 \( 1 + (-0.654 + 0.755i)T^{2} \)
73 \( 1 + (0.142 + 0.989i)T^{2} \)
79 \( 1 + (-0.258 + 1.80i)T + (-0.959 - 0.281i)T^{2} \)
83 \( 1 + (-0.415 + 0.909i)T^{2} \)
89 \( 1 + (1.84 - 0.540i)T + (0.841 - 0.540i)T^{2} \)
97 \( 1 + (-0.415 - 0.909i)T^{2} \)
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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

−9.356225275580860291842039675653, −8.848061322639479548719207053368, −7.76266478507750659527887788185, −7.33595376715233201689207732836, −6.82129951109202634947175070113, −4.51909018057337448954025778859, −4.02189972163525548459817766169, −3.19197036492448064429258159371, −2.18505487401638818558808968873, −1.35303133368798051177019477375, 2.04608667435664269152036601236, 2.86051475055511532780415747823, 3.91731601232397304558421219610, 4.95875537786533063104343341717, 5.80197678482934667547505127992, 6.83578934292064584682563097815, 7.938142060747297639536298179298, 8.510502013640778459804720491821, 8.624679519049711754216977210165, 9.632107505430983542454933572559

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