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Properties

Label 2-624-13.3-c1-0-3
Degree $2$
Conductor $624$
Sign $-0.396 - 0.918i$
Analytic cond. $4.98266$
Root an. cond. $2.23218$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)3-s + 0.561·5-s + (−1.78 + 3.08i)7-s + (−0.499 + 0.866i)9-s + (−1 − 1.73i)11-s + (0.5 + 3.57i)13-s + (0.280 + 0.486i)15-s + (−1.28 + 2.21i)17-s + (−0.561 + 0.972i)19-s − 3.56·21-s + (1 + 1.73i)23-s − 4.68·25-s − 0.999·27-s + (2.84 + 4.92i)29-s + 1.56·31-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + 0.251·5-s + (−0.673 + 1.16i)7-s + (−0.166 + 0.288i)9-s + (−0.301 − 0.522i)11-s + (0.138 + 0.990i)13-s + (0.0724 + 0.125i)15-s + (−0.310 + 0.538i)17-s + (−0.128 + 0.223i)19-s − 0.777·21-s + (0.208 + 0.361i)23-s − 0.936·25-s − 0.192·27-s + (0.527 + 0.914i)29-s + 0.280·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.396 - 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.396 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $-0.396 - 0.918i$
Analytic conductor: \(4.98266\)
Root analytic conductor: \(2.23218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{624} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 624,\ (\ :1/2),\ -0.396 - 0.918i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.709461 + 1.07879i\)
\(L(\frac12)\) \(\approx\) \(0.709461 + 1.07879i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (-0.5 - 3.57i)T \)
good5 \( 1 - 0.561T + 5T^{2} \)
7 \( 1 + (1.78 - 3.08i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.28 - 2.21i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.561 - 0.972i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1 - 1.73i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.84 - 4.92i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.56T + 31T^{2} \)
37 \( 1 + (1.71 + 2.97i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.28 + 2.21i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.219 + 0.379i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 8.24T + 47T^{2} \)
53 \( 1 - 11.6T + 53T^{2} \)
59 \( 1 + (5.56 - 9.63i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.06 - 10.4i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.219 - 0.379i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-7 + 12.1i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 1.87T + 73T^{2} \)
79 \( 1 + 9.56T + 79T^{2} \)
83 \( 1 - 9.12T + 83T^{2} \)
89 \( 1 + (6.56 + 11.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.21 + 3.84i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.74433477561483660809857766431, −9.959116886742986237879973981461, −8.877617502764620517177885350249, −8.794204341455004266217692018810, −7.35581226172507271576991438856, −6.16160295746653902693930145534, −5.56803360691824547869211221491, −4.26966843194268539304906899502, −3.15461203009852159155465827222, −2.06198272077028330074613126978, 0.66767711178970042718035025896, 2.39078032094598138211603057609, 3.52955931279805573222254118591, 4.68318435894283494086089971321, 5.97397455478970985323386903917, 6.88453650447518057245173773301, 7.59047933334261289274418968125, 8.459737355899943456000687419318, 9.681073733381650324333712081691, 10.17573819142272344005598591243

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