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Properties

Label 2-644-7.4-c3-0-18
Degree $2$
Conductor $644$
Sign $0.116 - 0.993i$
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  + (−3.96 + 6.87i)3-s + (−0.583 − 1.01i)5-s + (−10.1 + 15.5i)7-s + (−17.9 − 31.1i)9-s + (16.2 − 28.0i)11-s + 55.4·13-s + 9.25·15-s + (33.7 − 58.4i)17-s + (47.6 + 82.4i)19-s + (−66.5 − 130. i)21-s + (11.5 + 19.9i)23-s + (61.8 − 107. i)25-s + 70.8·27-s + 160.·29-s + (52.4 − 90.8i)31-s + ⋯
L(s)  = 1  + (−0.763 + 1.32i)3-s + (−0.0521 − 0.0903i)5-s + (−0.545 + 0.838i)7-s + (−0.665 − 1.15i)9-s + (0.444 − 0.769i)11-s + 1.18·13-s + 0.159·15-s + (0.481 − 0.833i)17-s + (0.575 + 0.995i)19-s + (−0.691 − 1.36i)21-s + (0.104 + 0.180i)23-s + (0.494 − 0.856i)25-s + 0.504·27-s + 1.02·29-s + (0.303 − 0.526i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.116 - 0.993i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.116 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(644\)    =    \(2^{2} \cdot 7 \cdot 23\)
Sign: $0.116 - 0.993i$
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: \((2,\ 644,\ (\ :3/2),\ 0.116 - 0.993i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.520747670\)
\(L(\frac12)\) \(\approx\) \(1.520747670\)
\(L(\frac{5}{2})\) 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 \( 1 + (10.1 - 15.5i)T \)
23 \( 1 + (-11.5 - 19.9i)T \)
good3 \( 1 + (3.96 - 6.87i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + (0.583 + 1.01i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-16.2 + 28.0i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 55.4T + 2.19e3T^{2} \)
17 \( 1 + (-33.7 + 58.4i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-47.6 - 82.4i)T + (-3.42e3 + 5.94e3i)T^{2} \)
29 \( 1 - 160.T + 2.43e4T^{2} \)
31 \( 1 + (-52.4 + 90.8i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (20.5 + 35.6i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 499.T + 6.89e4T^{2} \)
43 \( 1 + 95.3T + 7.95e4T^{2} \)
47 \( 1 + (-68.4 - 118. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (308. - 535. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (96.0 - 166. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (290. + 503. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-286. + 496. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 458.T + 3.57e5T^{2} \)
73 \( 1 + (-65.8 + 114. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-173. - 300. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 621.T + 5.71e5T^{2} \)
89 \( 1 + (-649. - 1.12e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 0.172T + 9.12e5T^{2} \)
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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

−10.39324260528896101285539474948, −9.502309984351071368132076308769, −8.969131455756391858985214441665, −7.925915903776480307820146658227, −6.24009763745083167131565196198, −5.91602137697524769205716307127, −4.88709181060038721171487181732, −3.83508198163988093081772171255, −2.96685634072761204039292869720, −0.851745011327057113690480209019, 0.75072609752614404669177216475, 1.51850414183534892473030975817, 3.16136494577768921866278645764, 4.40894851496642301479916717710, 5.71897744881358875545074064395, 6.60657330642305652463572281565, 7.04455647905733223528076729365, 7.928841639894208786194059302924, 9.033931649440461009106905560643, 10.15448282380539096347778598143

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