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Properties

Label 2-624-12.11-c3-0-11
Degree $2$
Conductor $624$
Sign $-0.711 - 0.702i$
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  + (−3.69 − 3.64i)3-s + 9.02i·5-s + 34.2i·7-s + (0.365 + 26.9i)9-s + 49.5·11-s + 13·13-s + (32.9 − 33.3i)15-s − 16.7i·17-s + 139. i·19-s + (124. − 126. i)21-s − 169.·23-s + 43.5·25-s + (97.1 − 101. i)27-s − 91.0i·29-s + 87.3i·31-s + ⋯
L(s)  = 1  + (−0.711 − 0.702i)3-s + 0.807i·5-s + 1.84i·7-s + (0.0135 + 0.999i)9-s + 1.35·11-s + 0.277·13-s + (0.566 − 0.574i)15-s − 0.239i·17-s + 1.68i·19-s + (1.29 − 1.31i)21-s − 1.53·23-s + 0.348·25-s + (0.692 − 0.721i)27-s − 0.583i·29-s + 0.506i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.711 - 0.702i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.711 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $-0.711 - 0.702i$
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: \((2,\ 624,\ (\ :3/2),\ -0.711 - 0.702i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.189495547\)
\(L(\frac12)\) \(\approx\) \(1.189495547\)
\(L(\frac{5}{2})\) 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 \( 1 + (3.69 + 3.64i)T \)
13 \( 1 - 13T \)
good5 \( 1 - 9.02iT - 125T^{2} \)
7 \( 1 - 34.2iT - 343T^{2} \)
11 \( 1 - 49.5T + 1.33e3T^{2} \)
17 \( 1 + 16.7iT - 4.91e3T^{2} \)
19 \( 1 - 139. iT - 6.85e3T^{2} \)
23 \( 1 + 169.T + 1.21e4T^{2} \)
29 \( 1 + 91.0iT - 2.43e4T^{2} \)
31 \( 1 - 87.3iT - 2.97e4T^{2} \)
37 \( 1 + 61.8T + 5.06e4T^{2} \)
41 \( 1 + 407. iT - 6.89e4T^{2} \)
43 \( 1 - 121. iT - 7.95e4T^{2} \)
47 \( 1 - 162.T + 1.03e5T^{2} \)
53 \( 1 - 504. iT - 1.48e5T^{2} \)
59 \( 1 + 796.T + 2.05e5T^{2} \)
61 \( 1 - 778.T + 2.26e5T^{2} \)
67 \( 1 - 481. iT - 3.00e5T^{2} \)
71 \( 1 - 965.T + 3.57e5T^{2} \)
73 \( 1 - 410.T + 3.89e5T^{2} \)
79 \( 1 - 193. iT - 4.93e5T^{2} \)
83 \( 1 + 530.T + 5.71e5T^{2} \)
89 \( 1 + 125. iT - 7.04e5T^{2} \)
97 \( 1 + 762.T + 9.12e5T^{2} \)
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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

−10.68745366034614583977169234065, −9.697632678440053494047858545225, −8.703165691574719179733775100205, −7.88279457984406951079232825816, −6.69107396577959783097517318900, −6.07538502399610753960331407656, −5.47008078837730386513649954442, −3.89488168173390498691640551958, −2.50468107952387305799609888449, −1.56865483738843084802451620822, 0.42107113713311545104531869067, 1.26269739502991782931477066852, 3.63101702566467419119125010613, 4.26369467507264382790657254949, 4.99311866564675562546288660257, 6.38521724656079321713948191547, 6.94560352595716234676415167415, 8.220947778829999640235792922664, 9.261634723013572327582383115446, 9.861079262122910103565199309504

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