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Properties

Label 2-624-13.10-c3-0-36
Degree $2$
Conductor $624$
Sign $-0.887 + 0.460i$
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 − 3.05i·5-s + (5.78 + 3.34i)7-s + (−4.5 + 7.79i)9-s + (27.9 − 16.1i)11-s + (−22.1 − 41.3i)13-s + (−7.92 + 4.57i)15-s + (−14.4 + 24.9i)17-s + (−87.6 − 50.5i)19-s − 20.0i·21-s + (59.4 + 103. i)23-s + 115.·25-s + 27·27-s + (−80.0 − 138. i)29-s + 38.0i·31-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s − 0.272i·5-s + (0.312 + 0.180i)7-s + (−0.166 + 0.288i)9-s + (0.765 − 0.441i)11-s + (−0.472 − 0.881i)13-s + (−0.136 + 0.0787i)15-s + (−0.205 + 0.356i)17-s + (−1.05 − 0.610i)19-s − 0.208i·21-s + (0.539 + 0.934i)23-s + 0.925·25-s + 0.192·27-s + (−0.512 − 0.887i)29-s + 0.220i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.887 + 0.460i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.887 + 0.460i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $-0.887 + 0.460i$
Analytic conductor: \(36.8171\)
Root analytic conductor: \(6.06771\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{624} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 624,\ (\ :3/2),\ -0.887 + 0.460i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.043446565\)
\(L(\frac12)\) \(\approx\) \(1.043446565\)
\(L(\frac{5}{2})\) 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 \( 1 + (22.1 + 41.3i)T \)
good5 \( 1 + 3.05iT - 125T^{2} \)
7 \( 1 + (-5.78 - 3.34i)T + (171.5 + 297. i)T^{2} \)
11 \( 1 + (-27.9 + 16.1i)T + (665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (14.4 - 24.9i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (87.6 + 50.5i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-59.4 - 103. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (80.0 + 138. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 38.0iT - 2.97e4T^{2} \)
37 \( 1 + (-283. + 163. i)T + (2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-48.5 + 28.0i)T + (3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (63.9 - 110. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + 517. iT - 1.03e5T^{2} \)
53 \( 1 + 695.T + 1.48e5T^{2} \)
59 \( 1 + (568. + 328. i)T + (1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (350. - 607. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-49.5 + 28.5i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (-267. - 154. i)T + (1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 - 389. iT - 3.89e5T^{2} \)
79 \( 1 + 901.T + 4.93e5T^{2} \)
83 \( 1 + 687. iT - 5.71e5T^{2} \)
89 \( 1 + (927. - 535. i)T + (3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (1.51e3 + 877. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−9.798391648709210839212791553267, −8.850893054322654307283771969522, −8.114322711144543883605150569996, −7.14190022871567284319012257428, −6.20763044287966289020636363344, −5.33019783553951861133507172674, −4.28807143859885581241637219007, −2.90467267165389145204946298721, −1.58673386576287674476291882419, −0.31694492745313527765715844477, 1.47943859755961317011008198114, 2.88998869506038267945463676740, 4.30096693633441185025309888438, 4.75259482738711401412838808948, 6.24266280318932134069161336094, 6.83050922333250271415968296044, 7.949647769143650579719615110350, 9.084832987196152268795686869278, 9.608959141439042252442505433958, 10.78263581099188830302957480274

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