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Properties

Label 2-630-63.58-c1-0-7
Degree $2$
Conductor $630$
Sign $-0.459 - 0.888i$
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  + 2-s + (0.184 + 1.72i)3-s + 4-s + (−0.5 − 0.866i)5-s + (0.184 + 1.72i)6-s + (−2.25 + 1.38i)7-s + 8-s + (−2.93 + 0.636i)9-s + (−0.5 − 0.866i)10-s + (−0.480 + 0.832i)11-s + (0.184 + 1.72i)12-s + (−2.94 + 5.09i)13-s + (−2.25 + 1.38i)14-s + (1.39 − 1.02i)15-s + 16-s + (3.89 + 6.75i)17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.106 + 0.994i)3-s + 0.5·4-s + (−0.223 − 0.387i)5-s + (0.0754 + 0.703i)6-s + (−0.853 + 0.521i)7-s + 0.353·8-s + (−0.977 + 0.212i)9-s + (−0.158 − 0.273i)10-s + (−0.144 + 0.251i)11-s + (0.0533 + 0.497i)12-s + (−0.816 + 1.41i)13-s + (−0.603 + 0.368i)14-s + (0.361 − 0.263i)15-s + 0.250·16-s + (0.945 + 1.63i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.459 - 0.888i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.459 - 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.459 - 0.888i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.459 - 0.888i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.904661 + 1.48581i\)
\(L(\frac12)\) \(\approx\) \(0.904661 + 1.48581i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (-0.184 - 1.72i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (2.25 - 1.38i)T \)
good11 \( 1 + (0.480 - 0.832i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.94 - 5.09i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.89 - 6.75i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.774 + 1.34i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.95 - 3.38i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.543 + 0.940i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 2.55T + 31T^{2} \)
37 \( 1 + (-5.84 + 10.1i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.15 + 1.99i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.18 + 3.79i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 3.55T + 47T^{2} \)
53 \( 1 + (-0.274 - 0.475i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 6.00T + 59T^{2} \)
61 \( 1 - 14.6T + 61T^{2} \)
67 \( 1 + 0.821T + 67T^{2} \)
71 \( 1 + 3.96T + 71T^{2} \)
73 \( 1 + (-0.368 - 0.638i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 6.79T + 79T^{2} \)
83 \( 1 + (-6.12 - 10.6i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.85 - 4.94i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.80 - 13.5i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.93376734295419343515533498893, −9.886823750268110180003826858988, −9.369116263076361730700362138063, −8.396875785536013588023278056024, −7.23162613817051203330405182775, −6.07703825585450740101748403761, −5.30122452454696558340248579281, −4.26266306132629945061625317403, −3.52200442462765191702375157145, −2.24892752690252664829405138641, 0.73178261100985527070782302072, 2.89078620225033700797945586114, 3.10568321747246964990529782146, 4.86455928277494741533686672605, 5.86771560875303269490697830828, 6.77654474824801063841277885004, 7.48580595442995166569341231644, 8.140582867277995561772444275055, 9.653930774845072064264119746218, 10.35801868914504383203637964491

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