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Properties

Label 2-624-13.9-c3-0-14
Degree $2$
Conductor $624$
Sign $0.927 - 0.372i$
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  + (−1.5 + 2.59i)3-s − 16.4·5-s + (4.83 + 8.38i)7-s + (−4.5 − 7.79i)9-s + (13.7 − 23.8i)11-s + (−37.3 + 28.3i)13-s + (24.6 − 42.6i)15-s + (−53.9 − 93.4i)17-s + (−1.12 − 1.94i)19-s − 29.0·21-s + (20.9 − 36.2i)23-s + 144.·25-s + 27·27-s + (−30.8 + 53.3i)29-s − 191.·31-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s − 1.46·5-s + (0.261 + 0.452i)7-s + (−0.166 − 0.288i)9-s + (0.378 − 0.654i)11-s + (−0.795 + 0.605i)13-s + (0.423 − 0.734i)15-s + (−0.769 − 1.33i)17-s + (−0.0135 − 0.0234i)19-s − 0.301·21-s + (0.189 − 0.328i)23-s + 1.15·25-s + 0.192·27-s + (−0.197 + 0.341i)29-s − 1.11·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 - 0.372i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.927 - 0.372i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $0.927 - 0.372i$
Analytic conductor: \(36.8171\)
Root analytic conductor: \(6.06771\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{624} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 624,\ (\ :3/2),\ 0.927 - 0.372i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9341232380\)
\(L(\frac12)\) \(\approx\) \(0.9341232380\)
\(L(\frac{5}{2})\) 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 \( 1 + (1.5 - 2.59i)T \)
13 \( 1 + (37.3 - 28.3i)T \)
good5 \( 1 + 16.4T + 125T^{2} \)
7 \( 1 + (-4.83 - 8.38i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-13.7 + 23.8i)T + (-665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (53.9 + 93.4i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (1.12 + 1.94i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-20.9 + 36.2i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (30.8 - 53.3i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + 191.T + 2.97e4T^{2} \)
37 \( 1 + (49.2 - 85.2i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-15.3 + 26.6i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-119. - 206. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 - 511.T + 1.03e5T^{2} \)
53 \( 1 - 492.T + 1.48e5T^{2} \)
59 \( 1 + (-242. - 419. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-222. - 384. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-95.0 + 164. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (-242. - 419. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + 957.T + 3.89e5T^{2} \)
79 \( 1 - 375.T + 4.93e5T^{2} \)
83 \( 1 - 715.T + 5.71e5T^{2} \)
89 \( 1 + (-519. + 899. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (32.7 + 56.7i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−10.42377005037791297377123149181, −9.126843793574910034799632087100, −8.748391353476305706128998248173, −7.51583241086890031761729317210, −6.88265250667275162052651025735, −5.50731009575257395516915659346, −4.57659147343354866415069818268, −3.80152691504847297416604826183, −2.57817519631466111597379913911, −0.55504434542636311598478314247, 0.57399395517810304400052009244, 2.08383052743919806309953661448, 3.70364657329080320035244865153, 4.38737296115553628309100587146, 5.57178157152381841731090638890, 6.89699574586073061277208293482, 7.47156496518268429609418493096, 8.141204882793390024975660303814, 9.145246505880889588700959267009, 10.47069090859791302922383127190

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