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Properties

Label 2-630-7.4-c3-0-24
Degree $2$
Conductor $630$
Sign $-0.503 + 0.864i$
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 + (−15.4 + 10.2i)7-s + 7.99·8-s + (5 − 8.66i)10-s + (−10.2 + 17.6i)11-s − 53.1·13-s + (33.1 + 16.5i)14-s + (−8 − 13.8i)16-s + (13.8 − 24.0i)17-s + (71.5 + 123. i)19-s − 20·20-s + 40.8·22-s + (−100. − 173. i)23-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.223 + 0.387i)5-s + (−0.834 + 0.550i)7-s + 0.353·8-s + (0.158 − 0.273i)10-s + (−0.280 + 0.485i)11-s − 1.13·13-s + (0.632 + 0.316i)14-s + (−0.125 − 0.216i)16-s + (0.197 − 0.342i)17-s + (0.864 + 1.49i)19-s − 0.223·20-s + 0.396·22-s + (−0.909 − 1.57i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.503 + 0.864i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.503 + 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.503 + 0.864i$
Analytic conductor: \(37.1712\)
Root analytic conductor: \(6.09681\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :3/2),\ -0.503 + 0.864i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.5931392374\)
\(L(\frac12)\) \(\approx\) \(0.5931392374\)
\(L(\frac{5}{2})\) 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 \( 1 + (15.4 - 10.2i)T \)
good11 \( 1 + (10.2 - 17.6i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 53.1T + 2.19e3T^{2} \)
17 \( 1 + (-13.8 + 24.0i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-71.5 - 123. i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (100. + 173. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 113.T + 2.43e4T^{2} \)
31 \( 1 + (-52.7 + 91.3i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-0.693 - 1.20i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 226.T + 6.89e4T^{2} \)
43 \( 1 - 268.T + 7.95e4T^{2} \)
47 \( 1 + (-13.8 - 24.0i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-37.1 + 64.3i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-332. + 576. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (254. + 441. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-490. + 849. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 144.T + 3.57e5T^{2} \)
73 \( 1 + (367. - 637. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (73.0 + 126. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 712.T + 5.71e5T^{2} \)
89 \( 1 + (303. + 525. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 771.T + 9.12e5T^{2} \)
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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

−9.793083219270830294253814705771, −9.487219814900050491677876002892, −8.169524638550002447519240118352, −7.39752685453478934838894578437, −6.31417898962868281249325404796, −5.32229085018168467241452102927, −4.02594460257835779463730080770, −2.85105343473094456712301020123, −2.06501766790359501038887057304, −0.22698970680040748300751602192, 0.986093139487749914497791083630, 2.70373648253202757246864497141, 4.02687288630563132512470753545, 5.22988023579874561264466165405, 5.96089159999895569558863706743, 7.17791100645579558739849526791, 7.59437580514474288831653253644, 8.865999924966339786673432407237, 9.565528336715296093219162359879, 10.15203330347167782825620554524

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