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Properties

Label 2-630-315.2-c1-0-31
Degree $2$
Conductor $630$
Sign $0.968 + 0.247i$
Analytic cond. $5.03057$
Root an. cond. $2.24289$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.965 − 0.258i)2-s + (1.54 + 0.781i)3-s + (0.866 + 0.499i)4-s + (1.53 − 1.62i)5-s + (−1.29 − 1.15i)6-s + (0.807 − 2.51i)7-s + (−0.707 − 0.707i)8-s + (1.77 + 2.41i)9-s + (−1.90 + 1.17i)10-s + 3.97i·11-s + (0.947 + 1.44i)12-s + (1.48 + 0.397i)13-s + (−1.43 + 2.22i)14-s + (3.64 − 1.32i)15-s + (0.500 + 0.866i)16-s + (6.68 + 1.79i)17-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s + (0.892 + 0.451i)3-s + (0.433 + 0.249i)4-s + (0.684 − 0.728i)5-s + (−0.526 − 0.471i)6-s + (0.305 − 0.952i)7-s + (−0.249 − 0.249i)8-s + (0.592 + 0.805i)9-s + (−0.601 + 0.372i)10-s + 1.19i·11-s + (0.273 + 0.418i)12-s + (0.411 + 0.110i)13-s + (−0.382 + 0.594i)14-s + (0.940 − 0.341i)15-s + (0.125 + 0.216i)16-s + (1.62 + 0.434i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.247i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 + 0.247i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.968 + 0.247i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (317, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.968 + 0.247i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.73812 - 0.218196i\)
\(L(\frac12)\) \(\approx\) \(1.73812 - 0.218196i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 + 0.258i)T \)
3 \( 1 + (-1.54 - 0.781i)T \)
5 \( 1 + (-1.53 + 1.62i)T \)
7 \( 1 + (-0.807 + 2.51i)T \)
good11 \( 1 - 3.97iT - 11T^{2} \)
13 \( 1 + (-1.48 - 0.397i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (-6.68 - 1.79i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (6.26 + 3.61i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.06 + 3.06i)T + 23iT^{2} \)
29 \( 1 + (2.71 - 4.70i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.95 + 8.57i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.509 + 0.136i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-2.17 + 1.25i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.68 - 6.27i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (-3.76 - 1.00i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (3.19 - 11.9i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-3.52 + 6.10i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.858 - 1.48i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.24 - 0.869i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 11.4iT - 71T^{2} \)
73 \( 1 + (6.17 + 1.65i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-1.77 + 1.02i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (13.9 - 3.74i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (7.41 - 12.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.76 - 0.742i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.22652728367617896425933255125, −9.761198251331600075532100526495, −8.916587310339313397750233312245, −8.071158241950779954545964084273, −7.41878152657632569125402916671, −6.17715815147001999311917151585, −4.69201036233337470268802732242, −4.01447883724901989825169904482, −2.43354959792644516138558420061, −1.37002624497277653999656857117, 1.51765682491768130761487328753, 2.62322418588867118770265859227, 3.54874750450990670279461044683, 5.68267870528647102073928859609, 6.12999121336018753575379669305, 7.26244949117240688535704352983, 8.280892323327833966080854162866, 8.629239513886639298716534673816, 9.728432008528451441475543821393, 10.31012092754365084819251815123

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