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Properties

Label 2-644-7.2-c3-0-2
Degree $2$
Conductor $644$
Sign $0.232 + 0.972i$
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  + (4.31 + 7.48i)3-s + (−8.44 + 14.6i)5-s + (−11.8 − 14.2i)7-s + (−23.8 + 41.2i)9-s + (0.473 + 0.820i)11-s − 78.8·13-s − 145.·15-s + (17.0 + 29.5i)17-s + (69.6 − 120. i)19-s + (55.2 − 150. i)21-s + (11.5 − 19.9i)23-s + (−80.1 − 138. i)25-s − 178.·27-s + 183.·29-s + (−58.1 − 100. i)31-s + ⋯
L(s)  = 1  + (0.831 + 1.43i)3-s + (−0.755 + 1.30i)5-s + (−0.640 − 0.768i)7-s + (−0.881 + 1.52i)9-s + (0.0129 + 0.0225i)11-s − 1.68·13-s − 2.51·15-s + (0.243 + 0.421i)17-s + (0.841 − 1.45i)19-s + (0.573 − 1.56i)21-s + (0.104 − 0.180i)23-s + (−0.641 − 1.11i)25-s − 1.27·27-s + 1.17·29-s + (−0.336 − 0.583i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.232 + 0.972i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.232 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(644\)    =    \(2^{2} \cdot 7 \cdot 23\)
Sign: $0.232 + 0.972i$
Analytic conductor: \(37.9972\)
Root analytic conductor: \(6.16418\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{644} (93, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 644,\ (\ :3/2),\ 0.232 + 0.972i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.08830696752\)
\(L(\frac12)\) \(\approx\) \(0.08830696752\)
\(L(\frac{5}{2})\) 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 \( 1 + (11.8 + 14.2i)T \)
23 \( 1 + (-11.5 + 19.9i)T \)
good3 \( 1 + (-4.31 - 7.48i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + (8.44 - 14.6i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-0.473 - 0.820i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 78.8T + 2.19e3T^{2} \)
17 \( 1 + (-17.0 - 29.5i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-69.6 + 120. i)T + (-3.42e3 - 5.94e3i)T^{2} \)
29 \( 1 - 183.T + 2.43e4T^{2} \)
31 \( 1 + (58.1 + 100. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (222. - 384. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 239.T + 6.89e4T^{2} \)
43 \( 1 + 268.T + 7.95e4T^{2} \)
47 \( 1 + (-18.9 + 32.8i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-110. - 191. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (402. + 696. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-41.5 + 71.9i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (251. + 435. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 1.01e3T + 3.57e5T^{2} \)
73 \( 1 + (-437. - 757. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (73.3 - 127. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 402.T + 5.71e5T^{2} \)
89 \( 1 + (526. - 911. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 786.T + 9.12e5T^{2} \)
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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

−10.60134964417872765443965593953, −9.951410296780333172749420545145, −9.471692024065978016702531500346, −8.237368755372396773720446973266, −7.33804726500237378604247015294, −6.70370717644901991968383416100, −4.97181134203432153483526501142, −4.21019056747862611031375420134, −3.13298120175153194928373923311, −2.80862260491634332339287280623, 0.02319067987455555328272755663, 1.23760064648620634526763199022, 2.42771762039526555094397313319, 3.46745579240735021722144771007, 4.92469714048735195382186135098, 5.88454244068570790732452739292, 7.22662800285847512054143989208, 7.64156899372820910657422398943, 8.577002655760879178766737687382, 9.119187383568589602354085448949

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