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Properties

Label 2-644-7.2-c3-0-19
Degree $2$
Conductor $644$
Sign $0.998 + 0.0483i$
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  + (−0.439 − 0.761i)3-s + (5.30 − 9.18i)5-s + (−16.9 + 7.41i)7-s + (13.1 − 22.7i)9-s + (29.3 + 50.8i)11-s + 39.3·13-s − 9.32·15-s + (−12.9 − 22.4i)17-s + (−67.1 + 116. i)19-s + (13.1 + 9.66i)21-s + (11.5 − 19.9i)23-s + (6.22 + 10.7i)25-s − 46.7·27-s − 21.6·29-s + (39.7 + 68.8i)31-s + ⋯
L(s)  = 1  + (−0.0845 − 0.146i)3-s + (0.474 − 0.821i)5-s + (−0.916 + 0.400i)7-s + (0.485 − 0.841i)9-s + (0.803 + 1.39i)11-s + 0.838·13-s − 0.160·15-s + (−0.184 − 0.320i)17-s + (−0.810 + 1.40i)19-s + (0.136 + 0.100i)21-s + (0.104 − 0.180i)23-s + (0.0497 + 0.0862i)25-s − 0.333·27-s − 0.138·29-s + (0.230 + 0.398i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0483i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0483i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(644\)    =    \(2^{2} \cdot 7 \cdot 23\)
Sign: $0.998 + 0.0483i$
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: \((2,\ 644,\ (\ :3/2),\ 0.998 + 0.0483i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.094250134\)
\(L(\frac12)\) \(\approx\) \(2.094250134\)
\(L(\frac{5}{2})\) 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 \( 1 + (16.9 - 7.41i)T \)
23 \( 1 + (-11.5 + 19.9i)T \)
good3 \( 1 + (0.439 + 0.761i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + (-5.30 + 9.18i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-29.3 - 50.8i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 39.3T + 2.19e3T^{2} \)
17 \( 1 + (12.9 + 22.4i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (67.1 - 116. i)T + (-3.42e3 - 5.94e3i)T^{2} \)
29 \( 1 + 21.6T + 2.43e4T^{2} \)
31 \( 1 + (-39.7 - 68.8i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-129. + 224. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 397.T + 6.89e4T^{2} \)
43 \( 1 - 259.T + 7.95e4T^{2} \)
47 \( 1 + (133. - 231. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-66.3 - 114. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (159. + 276. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-197. + 341. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (92.7 + 160. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 326.T + 3.57e5T^{2} \)
73 \( 1 + (-466. - 807. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-9.85 + 17.0i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 1.18e3T + 5.71e5T^{2} \)
89 \( 1 + (466. - 808. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 1.40e3T + 9.12e5T^{2} \)
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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

−9.778862728163895286814852631996, −9.428311218571001892679755856399, −8.688660161681389614157036164741, −7.38579658106607419581275686878, −6.41626840602674581827069420231, −5.86797616422068974732362056761, −4.48279185399452000630569855151, −3.67246647733789054042874637266, −2.03982763085511785812261928434, −0.967156803688908078909737300013, 0.801723525443906463549150872510, 2.44121667587437387486898860870, 3.46984441884377587788404835545, 4.46281817149434119458660622938, 6.02834391378579452968524995936, 6.39255148873075545720726019575, 7.36220686836115450663515222405, 8.574110299089341457224518187118, 9.335998580745920837400443201243, 10.38521603699726186704126579676

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