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Properties

Label 2-644-7.4-c3-0-9
Degree $2$
Conductor $644$
Sign $-0.990 + 0.140i$
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  + (−1.91 + 3.31i)3-s + (8.26 + 14.3i)5-s + (17.5 + 5.80i)7-s + (6.15 + 10.6i)9-s + (−22.5 + 39.0i)11-s − 80.5·13-s − 63.3·15-s + (−7.80 + 13.5i)17-s + (0.883 + 1.53i)19-s + (−52.9 + 47.2i)21-s + (11.5 + 19.9i)23-s + (−74.2 + 128. i)25-s − 150.·27-s + 137.·29-s + (12.7 − 22.1i)31-s + ⋯
L(s)  = 1  + (−0.368 + 0.638i)3-s + (0.739 + 1.28i)5-s + (0.949 + 0.313i)7-s + (0.228 + 0.394i)9-s + (−0.617 + 1.07i)11-s − 1.71·13-s − 1.09·15-s + (−0.111 + 0.192i)17-s + (0.0106 + 0.0184i)19-s + (−0.550 + 0.490i)21-s + (0.104 + 0.180i)23-s + (−0.594 + 1.02i)25-s − 1.07·27-s + 0.882·29-s + (0.0739 − 0.128i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.140i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.990 + 0.140i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(644\)    =    \(2^{2} \cdot 7 \cdot 23\)
Sign: $-0.990 + 0.140i$
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: \((2,\ 644,\ (\ :3/2),\ -0.990 + 0.140i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.589447133\)
\(L(\frac12)\) \(\approx\) \(1.589447133\)
\(L(\frac{5}{2})\) 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 \( 1 + (-17.5 - 5.80i)T \)
23 \( 1 + (-11.5 - 19.9i)T \)
good3 \( 1 + (1.91 - 3.31i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + (-8.26 - 14.3i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (22.5 - 39.0i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 80.5T + 2.19e3T^{2} \)
17 \( 1 + (7.80 - 13.5i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-0.883 - 1.53i)T + (-3.42e3 + 5.94e3i)T^{2} \)
29 \( 1 - 137.T + 2.43e4T^{2} \)
31 \( 1 + (-12.7 + 22.1i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (122. + 211. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 415.T + 6.89e4T^{2} \)
43 \( 1 - 293.T + 7.95e4T^{2} \)
47 \( 1 + (-30.5 - 52.9i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (124. - 214. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-185. + 322. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (131. + 227. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (280. - 485. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 683.T + 3.57e5T^{2} \)
73 \( 1 + (-109. + 189. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (360. + 624. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 205.T + 5.71e5T^{2} \)
89 \( 1 + (472. + 818. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 63.4T + 9.12e5T^{2} \)
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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

−10.44675906829419431857313397614, −10.06313955980133070666654988506, −9.194340611786845449048664210103, −7.59365251407754338094774292517, −7.33215535105611587160272351248, −5.97750036063774652533991528970, −5.05624784083485894736713874012, −4.40777680586724679281657097013, −2.61798322140395090543433994891, −2.04322726352161363792492090066, 0.47564490124088568302172449853, 1.32196792605403815959263342034, 2.56081090481099591305023051723, 4.42055610610055309563285290914, 5.16037973289087282007445336275, 5.92251421757138388257293501407, 7.11564174802102604781242964628, 7.949835158363759027282554999036, 8.803530624902635350850669106007, 9.654112908967829747130632080940

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