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Properties

Label 2-624-13.3-c3-0-34
Degree $2$
Conductor $624$
Sign $0.187 + 0.982i$
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 + 9.85·5-s + (14.9 − 25.9i)7-s + (−4.5 + 7.79i)9-s + (23.4 + 40.6i)11-s + (3.71 − 46.7i)13-s + (−14.7 − 25.5i)15-s + (24.1 − 41.7i)17-s + (60.1 − 104. i)19-s − 89.8·21-s + (65.3 + 113. i)23-s − 27.9·25-s + 27·27-s + (97.4 + 168. i)29-s + 32.0·31-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + 0.881·5-s + (0.808 − 1.40i)7-s + (−0.166 + 0.288i)9-s + (0.643 + 1.11i)11-s + (0.0791 − 0.996i)13-s + (−0.254 − 0.440i)15-s + (0.344 − 0.596i)17-s + (0.726 − 1.25i)19-s − 0.933·21-s + (0.592 + 1.02i)23-s − 0.223·25-s + 0.192·27-s + (0.624 + 1.08i)29-s + 0.185·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.187 + 0.982i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.187 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $0.187 + 0.982i$
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: \((2,\ 624,\ (\ :3/2),\ 0.187 + 0.982i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.533209433\)
\(L(\frac12)\) \(\approx\) \(2.533209433\)
\(L(\frac{5}{2})\) 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 \( 1 + (-3.71 + 46.7i)T \)
good5 \( 1 - 9.85T + 125T^{2} \)
7 \( 1 + (-14.9 + 25.9i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (-23.4 - 40.6i)T + (-665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (-24.1 + 41.7i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-60.1 + 104. i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-65.3 - 113. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-97.4 - 168. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 32.0T + 2.97e4T^{2} \)
37 \( 1 + (-16.2 - 28.0i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-120. - 209. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-48.2 + 83.4i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + 539.T + 1.03e5T^{2} \)
53 \( 1 + 152.T + 1.48e5T^{2} \)
59 \( 1 + (-163. + 283. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-49.2 + 85.2i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (220. + 382. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (-172. + 298. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 - 773.T + 3.89e5T^{2} \)
79 \( 1 - 150.T + 4.93e5T^{2} \)
83 \( 1 + 337.T + 5.71e5T^{2} \)
89 \( 1 + (84.9 + 147. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (107. - 185. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−9.980292808944841787908031834513, −9.391627563652964723747572335384, −7.993523648814864688284522272041, −7.27100140514707947002220671247, −6.64118364950848520669470941438, −5.28540937502586166819037678725, −4.68703562872724751773749429292, −3.18052840888050789229762847086, −1.66789895899830751625607151409, −0.863061728695205179557549492575, 1.35038626603412759449635626298, 2.46657787030549427364445442545, 3.84792678172945992339654935512, 5.05897041199400107313997381319, 5.91381363027876723396013406911, 6.34149465027814448795265353916, 8.077229950388786160041851546423, 8.773108434448558356495634524197, 9.476727191458512552267515238790, 10.32637468054430316554548526327

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