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Properties

Label 2-644-7.4-c3-0-10
Degree $2$
Conductor $644$
Sign $-0.963 + 0.266i$
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  + (−4.53 + 7.84i)3-s + (7.01 + 12.1i)5-s + (18.1 + 3.49i)7-s + (−27.5 − 47.7i)9-s + (16.5 − 28.6i)11-s + 38.7·13-s − 127.·15-s + (−54.7 + 94.8i)17-s + (37.1 + 64.3i)19-s + (−109. + 126. i)21-s + (11.5 + 19.9i)23-s + (−35.8 + 62.1i)25-s + 254.·27-s − 245.·29-s + (−117. + 204. i)31-s + ⋯
L(s)  = 1  + (−0.872 + 1.51i)3-s + (0.627 + 1.08i)5-s + (0.981 + 0.188i)7-s + (−1.02 − 1.76i)9-s + (0.453 − 0.786i)11-s + 0.827·13-s − 2.18·15-s + (−0.781 + 1.35i)17-s + (0.448 + 0.776i)19-s + (−1.14 + 1.31i)21-s + (0.104 + 0.180i)23-s + (−0.287 + 0.497i)25-s + 1.81·27-s − 1.56·29-s + (−0.682 + 1.18i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.963 + 0.266i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.963 + 0.266i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(644\)    =    \(2^{2} \cdot 7 \cdot 23\)
Sign: $-0.963 + 0.266i$
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: \((2,\ 644,\ (\ :3/2),\ -0.963 + 0.266i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.489305428\)
\(L(\frac12)\) \(\approx\) \(1.489305428\)
\(L(\frac{5}{2})\) 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 \( 1 + (-18.1 - 3.49i)T \)
23 \( 1 + (-11.5 - 19.9i)T \)
good3 \( 1 + (4.53 - 7.84i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + (-7.01 - 12.1i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-16.5 + 28.6i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 38.7T + 2.19e3T^{2} \)
17 \( 1 + (54.7 - 94.8i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-37.1 - 64.3i)T + (-3.42e3 + 5.94e3i)T^{2} \)
29 \( 1 + 245.T + 2.43e4T^{2} \)
31 \( 1 + (117. - 204. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-125. - 217. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 109.T + 6.89e4T^{2} \)
43 \( 1 + 380.T + 7.95e4T^{2} \)
47 \( 1 + (113. + 195. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-307. + 532. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (32.5 - 56.3i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-6.45 - 11.1i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (357. - 619. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 22.4T + 3.57e5T^{2} \)
73 \( 1 + (-3.41 + 5.91i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (23.3 + 40.4i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 664.T + 5.71e5T^{2} \)
89 \( 1 + (-297. - 514. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 79.5T + 9.12e5T^{2} \)
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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

−10.63473051415536696464484841213, −10.11218971886211268701789029800, −9.047573303918773824779199208987, −8.297225229951554345411732232420, −6.71303853517920206958196870204, −5.91558193998886161350877849852, −5.33554841288874783661353194131, −4.05306097935003694346231794845, −3.36447771909829969420528318932, −1.63171935650280849901877618499, 0.49382840035021853030420624954, 1.42405095637155156394827058958, 2.17284316992406406094043822708, 4.46426652157230864775494015787, 5.26344171838610284910467711121, 6.01746051646061153863364266785, 7.14286167171346052547696009340, 7.63040771876027647804147853911, 8.822142042326675918953339813259, 9.452640013701635918634318808255

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