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Properties

Label 2-624-13.10-c1-0-11
Degree $2$
Conductor $624$
Sign $-0.702 + 0.711i$
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 + 1.73i·5-s + (−2.36 − 1.36i)7-s + (−0.499 + 0.866i)9-s + (2.36 − 1.36i)11-s + (−2.59 − 2.5i)13-s + (1.49 − 0.866i)15-s + (−0.133 + 0.232i)17-s + (−4.09 − 2.36i)19-s + 2.73i·21-s + (−4.09 − 7.09i)23-s + 2.00·25-s + 0.999·27-s + (−3.96 − 6.86i)29-s + 1.46i·31-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + 0.774i·5-s + (−0.894 − 0.516i)7-s + (−0.166 + 0.288i)9-s + (0.713 − 0.411i)11-s + (−0.720 − 0.693i)13-s + (0.387 − 0.223i)15-s + (−0.0324 + 0.0562i)17-s + (−0.940 − 0.542i)19-s + 0.596i·21-s + (−0.854 − 1.48i)23-s + 0.400·25-s + 0.192·27-s + (−0.736 − 1.27i)29-s + 0.262i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.702 + 0.711i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.702 + 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $-0.702 + 0.711i$
Analytic conductor: \(4.98266\)
Root analytic conductor: \(2.23218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{624} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 624,\ (\ :1/2),\ -0.702 + 0.711i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.246608 - 0.590008i\)
\(L(\frac12)\) \(\approx\) \(0.246608 - 0.590008i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.59 + 2.5i)T \)
good5 \( 1 - 1.73iT - 5T^{2} \)
7 \( 1 + (2.36 + 1.36i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.36 + 1.36i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.133 - 0.232i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.09 + 2.36i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.09 + 7.09i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.96 + 6.86i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.46iT - 31T^{2} \)
37 \( 1 + (1.33 - 0.767i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.33 + 2.5i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (6.09 - 10.5i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 3.26iT - 47T^{2} \)
53 \( 1 + 7.92T + 53T^{2} \)
59 \( 1 + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.13 + 5.42i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.56 - 4.36i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.90 - 1.09i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 9.19iT - 73T^{2} \)
79 \( 1 - 8.39T + 79T^{2} \)
83 \( 1 - 1.66iT - 83T^{2} \)
89 \( 1 + (-8.19 + 4.73i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-8.66 - 5i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.38590870518700097398547750372, −9.592248039620659456019094620637, −8.421232905712610382205904581258, −7.47022779460204591282012657828, −6.50421675872121640982230222283, −6.21512284159051640079875331244, −4.64907809018882189331039822381, −3.43578924738439697636436439670, −2.36185824693527123645621071436, −0.34669280149948333190026072289, 1.83210314790305442979911372314, 3.47856226575362933397146902544, 4.43993941734565334596465131686, 5.43135843505391145629307869532, 6.33705870590852585922086579833, 7.30014986394718566548666685583, 8.621277338858645894607220649919, 9.351673085814520470856554935132, 9.787291450380876134619498073809, 10.91136363049323556264752960163

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