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Properties

Label 2-624-13.4-c3-0-35
Degree $2$
Conductor $624$
Sign $0.449 + 0.893i$
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 − 13.4i·5-s + (27.2 − 15.7i)7-s + (−4.5 − 7.79i)9-s + (35.0 + 20.2i)11-s + (42.1 − 20.5i)13-s + (34.9 + 20.1i)15-s + (−21.5 − 37.3i)17-s + (−23.3 + 13.4i)19-s + 94.2i·21-s + (9.50 − 16.4i)23-s − 55.6·25-s + 27·27-s + (77.0 − 133. i)29-s + 308. i·31-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s − 1.20i·5-s + (1.46 − 0.848i)7-s + (−0.166 − 0.288i)9-s + (0.961 + 0.555i)11-s + (0.898 − 0.438i)13-s + (0.601 + 0.347i)15-s + (−0.307 − 0.533i)17-s + (−0.282 + 0.162i)19-s + 0.979i·21-s + (0.0861 − 0.149i)23-s − 0.445·25-s + 0.192·27-s + (0.493 − 0.854i)29-s + 1.78i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.449 + 0.893i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.449 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $0.449 + 0.893i$
Analytic conductor: \(36.8171\)
Root analytic conductor: \(6.06771\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{624} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 624,\ (\ :3/2),\ 0.449 + 0.893i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.315427153\)
\(L(\frac12)\) \(\approx\) \(2.315427153\)
\(L(\frac{5}{2})\) 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 \( 1 + (-42.1 + 20.5i)T \)
good5 \( 1 + 13.4iT - 125T^{2} \)
7 \( 1 + (-27.2 + 15.7i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (-35.0 - 20.2i)T + (665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (21.5 + 37.3i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (23.3 - 13.4i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-9.50 + 16.4i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-77.0 + 133. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 - 308. iT - 2.97e4T^{2} \)
37 \( 1 + (37.6 + 21.7i)T + (2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-41.4 - 23.9i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (171. + 296. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + 133. iT - 1.03e5T^{2} \)
53 \( 1 + 438.T + 1.48e5T^{2} \)
59 \( 1 + (511. - 295. i)T + (1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-270. - 468. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-199. - 115. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (-389. + 224. i)T + (1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + 389. iT - 3.89e5T^{2} \)
79 \( 1 - 897.T + 4.93e5T^{2} \)
83 \( 1 + 1.30e3iT - 5.71e5T^{2} \)
89 \( 1 + (-801. - 462. i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-1.35e3 + 780. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−10.18194742782154684500429669633, −8.977081808097765262983779446532, −8.528897427085008604468465293612, −7.52663450977841387738523495479, −6.38550482995399343484340861158, −5.05856829400673491244323931857, −4.63882683338276902464176022366, −3.73370131308037102849231186591, −1.63993591260908823513394653948, −0.799253139610528296860670413823, 1.34625947031709002756158571867, 2.32473405457296153941293064067, 3.65489543680244543791826827888, 4.90783708885507170030794096301, 6.16762880284301229927690413633, 6.52486309847807929873428173794, 7.78985705574113492173449463510, 8.464145513766008106104076390394, 9.369151357077909036660755966042, 10.85485228341029859827585251011

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