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Properties

Label 2-630-315.292-c1-0-45
Degree $2$
Conductor $630$
Sign $-0.577 - 0.816i$
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  + (−0.707 − 0.707i)2-s + (−0.266 − 1.71i)3-s + 1.00i·4-s + (−0.634 + 2.14i)5-s + (−1.02 + 1.39i)6-s + (−1.85 − 1.88i)7-s + (0.707 − 0.707i)8-s + (−2.85 + 0.913i)9-s + (1.96 − 1.06i)10-s + (3.22 − 5.58i)11-s + (1.71 − 0.266i)12-s + (0.264 − 0.0707i)13-s + (−0.0185 + 2.64i)14-s + (3.83 + 0.514i)15-s − 1.00·16-s + (−1.85 − 0.496i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−0.154 − 0.988i)3-s + 0.500i·4-s + (−0.283 + 0.958i)5-s + (−0.416 + 0.571i)6-s + (−0.702 − 0.712i)7-s + (0.250 − 0.250i)8-s + (−0.952 + 0.304i)9-s + (0.621 − 0.337i)10-s + (0.972 − 1.68i)11-s + (0.494 − 0.0770i)12-s + (0.0732 − 0.0196i)13-s + (−0.00495 + 0.707i)14-s + (0.991 + 0.132i)15-s − 0.250·16-s + (−0.449 − 0.120i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.577 - 0.816i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.577 - 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0637386 + 0.123194i\)
\(L(\frac12)\) \(\approx\) \(0.0637386 + 0.123194i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 0.707i)T \)
3 \( 1 + (0.266 + 1.71i)T \)
5 \( 1 + (0.634 - 2.14i)T \)
7 \( 1 + (1.85 + 1.88i)T \)
good11 \( 1 + (-3.22 + 5.58i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.264 + 0.0707i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (1.85 + 0.496i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (3.64 - 6.31i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.00 + 1.07i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (6.46 - 3.73i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 8.30iT - 31T^{2} \)
37 \( 1 + (3.11 - 0.833i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (4.07 + 2.35i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.146 - 0.0391i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (-8.09 + 8.09i)T - 47iT^{2} \)
53 \( 1 + (7.07 + 1.89i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + 6.27T + 59T^{2} \)
61 \( 1 - 4.09iT - 61T^{2} \)
67 \( 1 + (4.73 + 4.73i)T + 67iT^{2} \)
71 \( 1 + 3.51T + 71T^{2} \)
73 \( 1 + (-0.669 + 2.50i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 - 3.29iT - 79T^{2} \)
83 \( 1 + (-2.02 + 7.55i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (-8.10 + 14.0i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.68 + 0.986i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.44489888527746830293781689495, −9.016910655451641029549743079038, −8.299557793460996303175933546611, −7.33975926585863891354583850686, −6.54859146319274947650366442080, −5.94928052946037262780796468953, −3.77212840535259122344562523493, −3.21549419706901592110618893216, −1.68524784105979378141626723653, −0.087768740540169061858952556408, 2.16420571417720090219678804624, 4.04820644718582954773042843176, 4.64473067754273392504765586567, 5.76511252317139033219663536261, 6.61342092955825357368001082154, 7.79934093758044465838074665418, 8.943060212366455792014018115009, 9.332099188830881826387941281066, 9.833032063935984001060101840150, 11.08749197012492152389694590161

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