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Properties

Label 2-644-644.195-c0-0-1
Degree $2$
Conductor $644$
Sign $0.603 - 0.797i$
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 + (−0.654 − 0.755i)7-s + (−0.142 + 0.989i)8-s + (0.959 + 0.281i)9-s + (0.239 + 0.153i)11-s + (−0.142 − 0.989i)14-s + (−0.654 + 0.755i)16-s + (0.654 + 0.755i)18-s + (0.118 + 0.258i)22-s + (−0.959 + 0.281i)23-s + (−0.841 + 0.540i)25-s + (0.415 − 0.909i)28-s + (−0.698 − 1.53i)29-s + (−0.959 + 0.281i)32-s + ⋯
L(s)  = 1  + (0.841 + 0.540i)2-s + (0.415 + 0.909i)4-s + (−0.654 − 0.755i)7-s + (−0.142 + 0.989i)8-s + (0.959 + 0.281i)9-s + (0.239 + 0.153i)11-s + (−0.142 − 0.989i)14-s + (−0.654 + 0.755i)16-s + (0.654 + 0.755i)18-s + (0.118 + 0.258i)22-s + (−0.959 + 0.281i)23-s + (−0.841 + 0.540i)25-s + (0.415 − 0.909i)28-s + (−0.698 − 1.53i)29-s + (−0.959 + 0.281i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.603 - 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.603 - 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(644\)    =    \(2^{2} \cdot 7 \cdot 23\)
Sign: $0.603 - 0.797i$
Analytic conductor: \(0.321397\)
Root analytic conductor: \(0.566919\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{644} (195, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 644,\ (\ :0),\ 0.603 - 0.797i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.426212687\)
\(L(\frac12)\) \(\approx\) \(1.426212687\)
\(L(1)\) 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 \( 1 + (0.654 + 0.755i)T \)
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

−10.98159086231213536672194083891, −10.04966148117687477186386969782, −9.222557338681238592414041832958, −7.78503816095351405743086324151, −7.38028101660177667425654918050, −6.40855376916982194746710992602, −5.53159195079931390018441952677, −4.18210096799366597931473876415, −3.78017519468288730713362677628, −2.13638851526052235030581545632, 1.70321731499122355163687785107, 3.02464921740565158653968290842, 3.99550467749531639950049967792, 5.02818209323460077533481917708, 6.16439433008079971530569168618, 6.67661229051591541463817605022, 7.982255642059685181861509489900, 9.346896719593310270676999816702, 9.790095331077183246732334318377, 10.72869189528752176476262576347

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