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Properties

Label 2-630-7.2-c3-0-27
Degree $2$
Conductor $630$
Sign $0.855 + 0.517i$
Analytic cond. $37.1712$
Root an. cond. $6.09681$
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 + 1.73i)2-s + (−1.99 − 3.46i)4-s + (2.5 − 4.33i)5-s + (18.4 + 1.54i)7-s + 7.99·8-s + (5 + 8.66i)10-s + (−23.7 − 41.1i)11-s + 1.12·13-s + (−21.1 + 30.4i)14-s + (−8 + 13.8i)16-s + (68.1 + 118. i)17-s + (44.4 − 76.9i)19-s − 20·20-s + 95.1·22-s + (1.36 − 2.36i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.223 − 0.387i)5-s + (0.996 + 0.0833i)7-s + 0.353·8-s + (0.158 + 0.273i)10-s + (−0.651 − 1.12i)11-s + 0.0240·13-s + (−0.403 + 0.580i)14-s + (−0.125 + 0.216i)16-s + (0.971 + 1.68i)17-s + (0.536 − 0.929i)19-s − 0.223·20-s + 0.921·22-s + (0.0123 − 0.0214i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.855 + 0.517i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.855 + 0.517i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.855 + 0.517i$
Analytic conductor: \(37.1712\)
Root analytic conductor: \(6.09681\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (541, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :3/2),\ 0.855 + 0.517i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.716804732\)
\(L(\frac12)\) \(\approx\) \(1.716804732\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 - 1.73i)T \)
3 \( 1 \)
5 \( 1 + (-2.5 + 4.33i)T \)
7 \( 1 + (-18.4 - 1.54i)T \)
good11 \( 1 + (23.7 + 41.1i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 1.12T + 2.19e3T^{2} \)
17 \( 1 + (-68.1 - 118. i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-44.4 + 76.9i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-1.36 + 2.36i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 131.T + 2.43e4T^{2} \)
31 \( 1 + (150. + 261. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (80.6 - 139. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 288.T + 6.89e4T^{2} \)
43 \( 1 + 70.5T + 7.95e4T^{2} \)
47 \( 1 + (-68.1 + 118. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (17.1 + 29.6i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (372. + 645. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-341. + 592. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-36.2 - 62.8i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 687.T + 3.57e5T^{2} \)
73 \( 1 + (42.2 + 73.1i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-361. + 625. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 1.01e3T + 5.71e5T^{2} \)
89 \( 1 + (-212. + 367. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 120.T + 9.12e5T^{2} \)
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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

−10.08953032980629439502399767862, −9.016720526929044958305714782459, −8.195548259068982471511972928437, −7.83849047164450767210755261289, −6.45209134089637474279409720932, −5.56163029960262627855632138839, −4.89708323418490771515546905996, −3.51377267168434018346304730027, −1.87018801758707376736240316086, −0.63464520845389298266009150197, 1.18082822464464510079884329932, 2.28148369331010962428347962254, 3.37819095491812068789021845044, 4.77229591167443215143080153732, 5.41716920876049017001468955815, 7.12168695737078869954048352736, 7.57895537834972868155767438545, 8.587373383814603989310882828845, 9.655468794282676459681409604588, 10.25105639534351013760849275763

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