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Properties

Label 2-644-7.2-c3-0-28
Degree $2$
Conductor $644$
Sign $0.721 - 0.691i$
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  + (3.73 + 6.46i)3-s + (5.71 − 9.89i)5-s + (−6.28 + 17.4i)7-s + (−14.3 + 24.8i)9-s + (−31.8 − 55.2i)11-s + 90.5·13-s + 85.2·15-s + (6.60 + 11.4i)17-s + (38.4 − 66.6i)19-s + (−136. + 24.3i)21-s + (11.5 − 19.9i)23-s + (−2.78 − 4.82i)25-s − 12.8·27-s + 123.·29-s + (167. + 290. i)31-s + ⋯
L(s)  = 1  + (0.718 + 1.24i)3-s + (0.511 − 0.885i)5-s + (−0.339 + 0.940i)7-s + (−0.531 + 0.921i)9-s + (−0.873 − 1.51i)11-s + 1.93·13-s + 1.46·15-s + (0.0941 + 0.163i)17-s + (0.464 − 0.804i)19-s + (−1.41 + 0.253i)21-s + (0.104 − 0.180i)23-s + (−0.0222 − 0.0385i)25-s − 0.0917·27-s + 0.793·29-s + (0.971 + 1.68i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.721 - 0.691i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.721 - 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(644\)    =    \(2^{2} \cdot 7 \cdot 23\)
Sign: $0.721 - 0.691i$
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: \((2,\ 644,\ (\ :3/2),\ 0.721 - 0.691i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.993221581\)
\(L(\frac12)\) \(\approx\) \(2.993221581\)
\(L(\frac{5}{2})\) 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 \( 1 + (6.28 - 17.4i)T \)
23 \( 1 + (-11.5 + 19.9i)T \)
good3 \( 1 + (-3.73 - 6.46i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + (-5.71 + 9.89i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (31.8 + 55.2i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 90.5T + 2.19e3T^{2} \)
17 \( 1 + (-6.60 - 11.4i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-38.4 + 66.6i)T + (-3.42e3 - 5.94e3i)T^{2} \)
29 \( 1 - 123.T + 2.43e4T^{2} \)
31 \( 1 + (-167. - 290. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (137. - 237. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 130.T + 6.89e4T^{2} \)
43 \( 1 - 39.5T + 7.95e4T^{2} \)
47 \( 1 + (-214. + 371. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-195. - 338. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (317. + 549. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-11.6 + 20.1i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (54.0 + 93.5i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 992.T + 3.57e5T^{2} \)
73 \( 1 + (-392. - 679. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (184. - 319. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 12.5T + 5.71e5T^{2} \)
89 \( 1 + (-497. + 862. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 1.66e3T + 9.12e5T^{2} \)
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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

−10.23377939993156179284597555080, −9.114726361658356761055422974437, −8.650328497462489177377131854291, −8.379545942929525384870743907475, −6.38204016811012966680504495335, −5.53299111351469493876530284810, −4.81270120118420277252536928255, −3.44406327837336270372398725643, −2.86872600203961538143339277047, −1.05631419098758160088314107229, 1.02970139874508137481808392878, 2.11490655775998077640078299380, 3.06471798199646270939255211766, 4.23238629753136923352030276806, 5.95559675559490819001027195645, 6.63856419368213783356329566898, 7.50713659138866784034363603251, 7.912742673243202523881208620285, 9.156597667265121387432602510532, 10.23220467112506224796023287735

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