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Properties

Label 2-624-12.11-c3-0-13
Degree $2$
Conductor $624$
Sign $0.757 - 0.653i$
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  + (−4.90 − 1.71i)3-s + 8.87i·5-s − 19.8i·7-s + (21.1 + 16.7i)9-s − 45.5·11-s + 13·13-s + (15.1 − 43.5i)15-s + 8.36i·17-s − 6.84i·19-s + (−33.8 + 97.1i)21-s − 44.6·23-s + 46.1·25-s + (−75.0 − 118. i)27-s − 194. i·29-s + 52.5i·31-s + ⋯
L(s)  = 1  + (−0.944 − 0.329i)3-s + 0.794i·5-s − 1.06i·7-s + (0.783 + 0.621i)9-s − 1.24·11-s + 0.277·13-s + (0.261 − 0.749i)15-s + 0.119i·17-s − 0.0826i·19-s + (−0.352 + 1.00i)21-s − 0.404·23-s + 0.369·25-s + (−0.534 − 0.845i)27-s − 1.24i·29-s + 0.304i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.757 - 0.653i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.757 - 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $0.757 - 0.653i$
Analytic conductor: \(36.8171\)
Root analytic conductor: \(6.06771\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{624} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 624,\ (\ :3/2),\ 0.757 - 0.653i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9646296702\)
\(L(\frac12)\) \(\approx\) \(0.9646296702\)
\(L(\frac{5}{2})\) 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 \( 1 + (4.90 + 1.71i)T \)
13 \( 1 - 13T \)
good5 \( 1 - 8.87iT - 125T^{2} \)
7 \( 1 + 19.8iT - 343T^{2} \)
11 \( 1 + 45.5T + 1.33e3T^{2} \)
17 \( 1 - 8.36iT - 4.91e3T^{2} \)
19 \( 1 + 6.84iT - 6.85e3T^{2} \)
23 \( 1 + 44.6T + 1.21e4T^{2} \)
29 \( 1 + 194. iT - 2.43e4T^{2} \)
31 \( 1 - 52.5iT - 2.97e4T^{2} \)
37 \( 1 + 439.T + 5.06e4T^{2} \)
41 \( 1 - 387. iT - 6.89e4T^{2} \)
43 \( 1 + 209. iT - 7.95e4T^{2} \)
47 \( 1 - 394.T + 1.03e5T^{2} \)
53 \( 1 - 412. iT - 1.48e5T^{2} \)
59 \( 1 + 191.T + 2.05e5T^{2} \)
61 \( 1 - 400.T + 2.26e5T^{2} \)
67 \( 1 - 803. iT - 3.00e5T^{2} \)
71 \( 1 - 687.T + 3.57e5T^{2} \)
73 \( 1 - 600.T + 3.89e5T^{2} \)
79 \( 1 - 1.20e3iT - 4.93e5T^{2} \)
83 \( 1 - 932.T + 5.71e5T^{2} \)
89 \( 1 - 282. iT - 7.04e5T^{2} \)
97 \( 1 - 1.30e3T + 9.12e5T^{2} \)
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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

−10.57060393157294858206392020953, −9.876261425521153070317481468040, −8.275822543731644737775086165047, −7.41382652987799808903200346396, −6.82193591338284675818716421155, −5.85463266207415708114114942904, −4.85914191507896913013484350951, −3.74337527936526648998325969977, −2.34845097181981869352046250093, −0.808709866336358899331404958531, 0.44688309429066602562924754024, 1.99355885309266305985493007559, 3.52094581993288386900800912684, 5.00867395627984654303599061273, 5.26396844740688979389903333529, 6.25497867410190088229309721754, 7.42228998046758780409570377268, 8.585929050522846902525465870060, 9.162315632215524781161969110068, 10.27951680716402442556214085127

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