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Properties

Label 2-624-13.12-c3-0-34
Degree $2$
Conductor $624$
Sign $-0.719 + 0.694i$
Analytic cond. $36.8171$
Root an. cond. $6.06771$
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  − 3·3-s − 12.9i·5-s − 16.7i·7-s + 9·9-s − 24.9i·11-s + (33.7 − 32.5i)13-s + 38.7i·15-s + 134.·17-s − 14.9i·19-s + 50.1i·21-s + 72·23-s − 41.7·25-s − 27·27-s − 206.·29-s − 249. i·31-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.15i·5-s − 0.903i·7-s + 0.333·9-s − 0.683i·11-s + (0.719 − 0.694i)13-s + 0.666i·15-s + 1.91·17-s − 0.180i·19-s + 0.521i·21-s + 0.652·23-s − 0.333·25-s − 0.192·27-s − 1.32·29-s − 1.44i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.719 + 0.694i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.719 + 0.694i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $-0.719 + 0.694i$
Analytic conductor: \(36.8171\)
Root analytic conductor: \(6.06771\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{624} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 624,\ (\ :3/2),\ -0.719 + 0.694i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.627895225\)
\(L(\frac12)\) \(\approx\) \(1.627895225\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + 3T \)
13 \( 1 + (-33.7 + 32.5i)T \)
good5 \( 1 + 12.9iT - 125T^{2} \)
7 \( 1 + 16.7iT - 343T^{2} \)
11 \( 1 + 24.9iT - 1.33e3T^{2} \)
17 \( 1 - 134.T + 4.91e3T^{2} \)
19 \( 1 + 14.9iT - 6.85e3T^{2} \)
23 \( 1 - 72T + 1.21e4T^{2} \)
29 \( 1 + 206.T + 2.43e4T^{2} \)
31 \( 1 + 249. iT - 2.97e4T^{2} \)
37 \( 1 - 293. iT - 5.06e4T^{2} \)
41 \( 1 + 250. iT - 6.89e4T^{2} \)
43 \( 1 - 432.T + 7.95e4T^{2} \)
47 \( 1 - 159. iT - 1.03e5T^{2} \)
53 \( 1 + 194.T + 1.48e5T^{2} \)
59 \( 1 + 232. iT - 2.05e5T^{2} \)
61 \( 1 + 185.T + 2.26e5T^{2} \)
67 \( 1 + 39.4iT - 3.00e5T^{2} \)
71 \( 1 - 920. iT - 3.57e5T^{2} \)
73 \( 1 + 549. iT - 3.89e5T^{2} \)
79 \( 1 + 933.T + 4.93e5T^{2} \)
83 \( 1 + 1.09e3iT - 5.71e5T^{2} \)
89 \( 1 + 532. iT - 7.04e5T^{2} \)
97 \( 1 - 362. iT - 9.12e5T^{2} \)
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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

−9.942232480393558322543401371244, −9.039132614742320386394587237849, −8.034341547883292948282007435509, −7.37569669523821019288027810765, −5.94648836305101518383354422432, −5.41845105668020644916392305368, −4.29750042132012070372497520901, −3.32190541622687834978804157490, −1.21555590437516458524483464041, −0.61200891715028187028052419052, 1.47473747671249465120200783535, 2.80191696755679826239559333043, 3.85403956890721691057494654794, 5.28453503870912372585151838280, 5.99173331311633127457075692325, 6.95186559264731397693035766090, 7.65328486691463083068989790277, 8.948744185091235232737982203361, 9.760264907466072778174010828736, 10.67192835032210166218988965753

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