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Properties

Label 2-624-13.3-c3-0-0
Degree $2$
Conductor $624$
Sign $-0.872 + 0.488i$
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 + 7·5-s + (−5 + 8.66i)7-s + (−4.5 + 7.79i)9-s + (−11 − 19.0i)11-s + (−45.5 + 11.2i)13-s + (10.5 + 18.1i)15-s + (−18.5 + 32.0i)17-s + (15 − 25.9i)19-s − 30.0·21-s + (−81 − 140. i)23-s − 76·25-s − 27·27-s + (56.5 + 97.8i)29-s − 196·31-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + 0.626·5-s + (−0.269 + 0.467i)7-s + (−0.166 + 0.288i)9-s + (−0.301 − 0.522i)11-s + (−0.970 + 0.240i)13-s + (0.180 + 0.313i)15-s + (−0.263 + 0.457i)17-s + (0.181 − 0.313i)19-s − 0.311·21-s + (−0.734 − 1.27i)23-s − 0.607·25-s − 0.192·27-s + (0.361 + 0.626i)29-s − 1.13·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.872 + 0.488i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.872 + 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $-0.872 + 0.488i$
Analytic conductor: \(36.8171\)
Root analytic conductor: \(6.06771\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{624} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 624,\ (\ :3/2),\ -0.872 + 0.488i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.06927072486\)
\(L(\frac12)\) \(\approx\) \(0.06927072486\)
\(L(\frac{5}{2})\) 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 \( 1 + (45.5 - 11.2i)T \)
good5 \( 1 - 7T + 125T^{2} \)
7 \( 1 + (5 - 8.66i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (11 + 19.0i)T + (-665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (18.5 - 32.0i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-15 + 25.9i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (81 + 140. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-56.5 - 97.8i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 196T + 2.97e4T^{2} \)
37 \( 1 + (6.5 + 11.2i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (142.5 + 246. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (123 - 213. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 462T + 1.03e5T^{2} \)
53 \( 1 + 537T + 1.48e5T^{2} \)
59 \( 1 + (-288 + 498. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-317.5 + 549. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-101 - 174. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (543 - 940. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + 805T + 3.89e5T^{2} \)
79 \( 1 + 884T + 4.93e5T^{2} \)
83 \( 1 + 518T + 5.71e5T^{2} \)
89 \( 1 + (97 + 168. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-601 + 1.04e3i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−10.49950036716132014414515089839, −9.848957615391255238600726546009, −9.035840374928631461841546074396, −8.317434358778121964977440262825, −7.15296313469166780666282008677, −6.07482470278223436915330007533, −5.27690259441214228639905394875, −4.19665883146835978392188446249, −2.91606373878424086424647769692, −2.00143037833824063966941124340, 0.01716246547149166287698131723, 1.64007005488730284389196547953, 2.63800195760024291637682358411, 3.91586279084517783116866237641, 5.19920515309984855740661516888, 6.08469666697605443331302708344, 7.26722523541226245238527692501, 7.64130173056891933213640918852, 8.920145023610678355236830526787, 9.840497167331731853830892298358

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