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Properties

Label 2-644-7.4-c3-0-6
Degree $2$
Conductor $644$
Sign $-0.883 - 0.468i$
Analytic cond. $37.9972$
Root an. cond. $6.16418$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.77 + 3.06i)3-s + (−3.18 − 5.51i)5-s + (10.8 + 15.0i)7-s + (7.22 + 12.5i)9-s + (−17.6 + 30.5i)11-s + 21.5·13-s + 22.5·15-s + (−9.63 + 16.6i)17-s + (−13.4 − 23.2i)19-s + (−65.2 + 6.49i)21-s + (11.5 + 19.9i)23-s + (42.2 − 73.1i)25-s − 146.·27-s + 242.·29-s + (27.3 − 47.3i)31-s + ⋯
L(s)  = 1  + (−0.340 + 0.590i)3-s + (−0.284 − 0.493i)5-s + (0.583 + 0.812i)7-s + (0.267 + 0.463i)9-s + (−0.484 + 0.838i)11-s + 0.458·13-s + 0.388·15-s + (−0.137 + 0.238i)17-s + (−0.162 − 0.281i)19-s + (−0.678 + 0.0674i)21-s + (0.104 + 0.180i)23-s + (0.337 − 0.585i)25-s − 1.04·27-s + 1.55·29-s + (0.158 − 0.274i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.883 - 0.468i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.883 - 0.468i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(644\)    =    \(2^{2} \cdot 7 \cdot 23\)
Sign: $-0.883 - 0.468i$
Analytic conductor: \(37.9972\)
Root analytic conductor: \(6.16418\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{644} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 644,\ (\ :3/2),\ -0.883 - 0.468i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.196534816\)
\(L(\frac12)\) \(\approx\) \(1.196534816\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-10.8 - 15.0i)T \)
23 \( 1 + (-11.5 - 19.9i)T \)
good3 \( 1 + (1.77 - 3.06i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + (3.18 + 5.51i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (17.6 - 30.5i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 21.5T + 2.19e3T^{2} \)
17 \( 1 + (9.63 - 16.6i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (13.4 + 23.2i)T + (-3.42e3 + 5.94e3i)T^{2} \)
29 \( 1 - 242.T + 2.43e4T^{2} \)
31 \( 1 + (-27.3 + 47.3i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-189. - 327. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 323.T + 6.89e4T^{2} \)
43 \( 1 + 489.T + 7.95e4T^{2} \)
47 \( 1 + (-106. - 185. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-155. + 269. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (166. - 288. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-276. - 479. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (98.5 - 170. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 1.09e3T + 3.57e5T^{2} \)
73 \( 1 + (163. - 283. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (127. + 221. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 956.T + 5.71e5T^{2} \)
89 \( 1 + (486. + 843. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 582.T + 9.12e5T^{2} \)
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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

−10.38383634582322749007158242851, −9.885951097041257584550873937208, −8.594046631376248907144502952026, −8.212987502467256146705716164159, −6.99737184509520004029490262378, −5.81201898399847148596913315909, −4.77559173606139158582733653106, −4.47344165657269695461137215286, −2.76993325434891786084323607439, −1.49229343997682233726259207703, 0.37768602090826494861496795997, 1.45091994144112039675394712145, 3.07681195073488549520410201490, 4.07332581934540353483574972792, 5.28549272158040494143145114909, 6.42715462224256975751643611988, 7.04440855490125525374237364279, 7.937405705754495421781085573506, 8.726376253122551411475193043716, 10.03446406646474728126274073336

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