Properties

Label 2-630-315.2-c1-0-35
Degree $2$
Conductor $630$
Sign $0.497 + 0.867i$
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 + (1.59 − 0.681i)3-s + (0.866 + 0.499i)4-s + (1.93 − 1.11i)5-s + (−1.71 + 0.246i)6-s + (1.44 + 2.21i)7-s + (−0.707 − 0.707i)8-s + (2.07 − 2.17i)9-s + (−2.16 + 0.578i)10-s − 5.61i·11-s + (1.71 + 0.206i)12-s + (−5.23 − 1.40i)13-s + (−0.825 − 2.51i)14-s + (2.32 − 3.09i)15-s + (0.500 + 0.866i)16-s + (3.42 + 0.917i)17-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s + (0.919 − 0.393i)3-s + (0.433 + 0.249i)4-s + (0.866 − 0.499i)5-s + (−0.699 + 0.100i)6-s + (0.547 + 0.836i)7-s + (−0.249 − 0.249i)8-s + (0.690 − 0.723i)9-s + (−0.683 + 0.182i)10-s − 1.69i·11-s + (0.496 + 0.0594i)12-s + (−1.45 − 0.388i)13-s + (−0.220 − 0.671i)14-s + (0.599 − 0.800i)15-s + (0.125 + 0.216i)16-s + (0.830 + 0.222i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.497 + 0.867i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.497 + 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.497 + 0.867i$
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: \((2,\ 630,\ (\ :1/2),\ 0.497 + 0.867i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.51832 - 0.879217i\)
\(L(\frac12)\) \(\approx\) \(1.51832 - 0.879217i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.59 + 0.681i)T \)
5 \( 1 + (-1.93 + 1.11i)T \)
7 \( 1 + (-1.44 - 2.21i)T \)
good11 \( 1 + 5.61iT - 11T^{2} \)
13 \( 1 + (5.23 + 1.40i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (-3.42 - 0.917i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (2.96 + 1.71i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.84 - 4.84i)T + 23iT^{2} \)
29 \( 1 + (3.11 - 5.39i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.92 - 5.05i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-6.67 + 1.78i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (0.460 - 0.265i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.05 + 3.92i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (-0.112 - 0.0301i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (0.378 - 1.41i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (1.40 - 2.42i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.53 - 7.85i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.47 - 0.663i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 9.18iT - 71T^{2} \)
73 \( 1 + (-5.01 - 1.34i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (13.3 - 7.70i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-14.0 + 3.76i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (-2.54 + 4.41i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.96 - 0.527i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.24232940868354828310973592825, −9.220486872417274442886122804152, −8.883861478140907874505312255687, −8.084142134033288355982976908648, −7.19366628895262767136494547548, −5.93279217456256383711850493076, −5.11171476880514074664261649711, −3.26761894815610496439310202173, −2.39898075451748945592218740942, −1.20473156082364606618208278467, 1.82628708661972877935996929277, 2.58007678202792551056226347362, 4.25663134236081795320634212754, 5.07964112463922343380890181000, 6.69206040101731625300945166988, 7.39735672985974320485387411091, 7.961400959823130439645928673404, 9.346166711934962942681268064262, 9.804632987740454137075151434680, 10.26887439906821237782921256195

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