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Properties

Label 2-644-644.111-c0-0-1
Degree $2$
Conductor $644$
Sign $-0.529 + 0.848i$
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 + (−0.142 − 0.989i)7-s + (−0.959 + 0.281i)8-s + (−0.841 + 0.540i)9-s + (0.797 − 1.74i)11-s + (−0.959 − 0.281i)14-s + (−0.142 + 0.989i)16-s + (0.142 + 0.989i)18-s + (−1.25 − 1.45i)22-s + (0.841 + 0.540i)23-s + (−0.415 − 0.909i)25-s + (−0.654 + 0.755i)28-s + (0.544 + 0.627i)29-s + (0.841 + 0.540i)32-s + ⋯
L(s)  = 1  + (0.415 − 0.909i)2-s + (−0.654 − 0.755i)4-s + (−0.142 − 0.989i)7-s + (−0.959 + 0.281i)8-s + (−0.841 + 0.540i)9-s + (0.797 − 1.74i)11-s + (−0.959 − 0.281i)14-s + (−0.142 + 0.989i)16-s + (0.142 + 0.989i)18-s + (−1.25 − 1.45i)22-s + (0.841 + 0.540i)23-s + (−0.415 − 0.909i)25-s + (−0.654 + 0.755i)28-s + (0.544 + 0.627i)29-s + (0.841 + 0.540i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.529 + 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.529 + 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(644\)    =    \(2^{2} \cdot 7 \cdot 23\)
Sign: $-0.529 + 0.848i$
Analytic conductor: \(0.321397\)
Root analytic conductor: \(0.566919\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{644} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 644,\ (\ :0),\ -0.529 + 0.848i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9850051612\)
\(L(\frac12)\) \(\approx\) \(0.9850051612\)
\(L(1)\) 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 \( 1 + (0.142 + 0.989i)T \)
23 \( 1 + (-0.841 - 0.540i)T \)
good3 \( 1 + (0.841 - 0.540i)T^{2} \)
5 \( 1 + (0.415 + 0.909i)T^{2} \)
11 \( 1 + (-0.797 + 1.74i)T + (-0.654 - 0.755i)T^{2} \)
13 \( 1 + (0.959 + 0.281i)T^{2} \)
17 \( 1 + (-0.142 + 0.989i)T^{2} \)
19 \( 1 + (0.142 + 0.989i)T^{2} \)
29 \( 1 + (-0.544 - 0.627i)T + (-0.142 + 0.989i)T^{2} \)
31 \( 1 + (0.841 + 0.540i)T^{2} \)
37 \( 1 + (-1.07 - 1.66i)T + (-0.415 + 0.909i)T^{2} \)
41 \( 1 + (-0.415 - 0.909i)T^{2} \)
43 \( 1 + (1.61 - 0.474i)T + (0.841 - 0.540i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (-1.80 + 0.258i)T + (0.959 - 0.281i)T^{2} \)
59 \( 1 + (-0.959 - 0.281i)T^{2} \)
61 \( 1 + (0.841 + 0.540i)T^{2} \)
67 \( 1 + (-0.544 - 1.19i)T + (-0.654 + 0.755i)T^{2} \)
71 \( 1 + (0.512 - 0.234i)T + (0.654 - 0.755i)T^{2} \)
73 \( 1 + (0.142 + 0.989i)T^{2} \)
79 \( 1 + (-0.118 + 0.822i)T + (-0.959 - 0.281i)T^{2} \)
83 \( 1 + (-0.415 + 0.909i)T^{2} \)
89 \( 1 + (0.841 - 0.540i)T^{2} \)
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

−10.65005074931358341455777410872, −9.891580955553663883073734294284, −8.768929181933916847063220236839, −8.198153462684166395386053044625, −6.66510200925045803854925561058, −5.83180212842871502472985087568, −4.77824658379035188934302503157, −3.64404626424773992238246294477, −2.88183659367968222508277016615, −1.08004331203658287825591957941, 2.42369779876652084127450154494, 3.71618036351885657688916811097, 4.81140634409601470531903045501, 5.72523856898641476924216991420, 6.57937541216715488396747831001, 7.35228117955439277874615288438, 8.495904321791663914227333439117, 9.191632722208906186421657152727, 9.776890057750255362141784989751, 11.38996316291292910691842521835

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