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Properties

Label 2-624-156.155-c3-0-59
Degree $2$
Conductor $624$
Sign $-0.313 + 0.949i$
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  + (2.74 − 4.41i)3-s − 14.3·5-s + 18.8·7-s + (−11.9 − 24.2i)9-s + 26.9i·11-s + (45.5 − 11.0i)13-s + (−39.2 + 63.1i)15-s − 26.8i·17-s + 108.·19-s + (51.7 − 83.3i)21-s + 14.6·23-s + 79.6·25-s + (−139. − 13.6i)27-s − 249. i·29-s + 29.9·31-s + ⋯
L(s)  = 1  + (0.527 − 0.849i)3-s − 1.27·5-s + 1.01·7-s + (−0.442 − 0.896i)9-s + 0.739i·11-s + (0.971 − 0.235i)13-s + (−0.675 + 1.08i)15-s − 0.383i·17-s + 1.30·19-s + (0.538 − 0.865i)21-s + 0.132·23-s + 0.636·25-s + (−0.995 − 0.0975i)27-s − 1.59i·29-s + 0.173·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.313 + 0.949i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.313 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $-0.313 + 0.949i$
Analytic conductor: \(36.8171\)
Root analytic conductor: \(6.06771\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{624} (623, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 624,\ (\ :3/2),\ -0.313 + 0.949i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.936785657\)
\(L(\frac12)\) \(\approx\) \(1.936785657\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.74 + 4.41i)T \)
13 \( 1 + (-45.5 + 11.0i)T \)
good5 \( 1 + 14.3T + 125T^{2} \)
7 \( 1 - 18.8T + 343T^{2} \)
11 \( 1 - 26.9iT - 1.33e3T^{2} \)
17 \( 1 + 26.8iT - 4.91e3T^{2} \)
19 \( 1 - 108.T + 6.85e3T^{2} \)
23 \( 1 - 14.6T + 1.21e4T^{2} \)
29 \( 1 + 249. iT - 2.43e4T^{2} \)
31 \( 1 - 29.9T + 2.97e4T^{2} \)
37 \( 1 - 214. iT - 5.06e4T^{2} \)
41 \( 1 + 177.T + 6.89e4T^{2} \)
43 \( 1 + 319. iT - 7.95e4T^{2} \)
47 \( 1 + 394. iT - 1.03e5T^{2} \)
53 \( 1 + 130. iT - 1.48e5T^{2} \)
59 \( 1 - 38.7iT - 2.05e5T^{2} \)
61 \( 1 - 25.9T + 2.26e5T^{2} \)
67 \( 1 - 56.9T + 3.00e5T^{2} \)
71 \( 1 + 382. iT - 3.57e5T^{2} \)
73 \( 1 + 1.03e3iT - 3.89e5T^{2} \)
79 \( 1 + 760. iT - 4.93e5T^{2} \)
83 \( 1 - 1.01e3iT - 5.71e5T^{2} \)
89 \( 1 + 300.T + 7.04e5T^{2} \)
97 \( 1 - 1.29e3iT - 9.12e5T^{2} \)
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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

−9.860059490654456830535454933733, −8.734643211996294791687647527571, −7.996065601328193939705826761719, −7.57339081993680998504270766168, −6.64371546739873035216796056525, −5.29107050678569012659474570942, −4.14538771654084252532574507450, −3.19346436041413942098093989680, −1.78421533528202334528178678246, −0.59099503578053225949217778916, 1.27768807193348317812528712135, 3.08871065296493336216313355226, 3.82609808546449531808289303299, 4.72042246311266793270602800364, 5.68356037974025205586809858757, 7.23211411704335145587399724800, 8.127885537476083401227854697983, 8.519650169153804339905725396540, 9.458946024171463835305518010583, 10.76128542154017985113160861709

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