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Properties

Label 2-630-63.25-c1-0-23
Degree $2$
Conductor $630$
Sign $0.136 + 0.990i$
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 + (0.681 + 1.59i)3-s + 4-s + (0.5 − 0.866i)5-s + (−0.681 − 1.59i)6-s + (−1.52 − 2.16i)7-s − 8-s + (−2.07 + 2.16i)9-s + (−0.5 + 0.866i)10-s + (−0.821 − 1.42i)11-s + (0.681 + 1.59i)12-s + (−2.49 − 4.31i)13-s + (1.52 + 2.16i)14-s + (1.71 + 0.206i)15-s + 16-s + (3.52 − 6.10i)17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.393 + 0.919i)3-s + 0.5·4-s + (0.223 − 0.387i)5-s + (−0.278 − 0.650i)6-s + (−0.575 − 0.817i)7-s − 0.353·8-s + (−0.690 + 0.723i)9-s + (−0.158 + 0.273i)10-s + (−0.247 − 0.429i)11-s + (0.196 + 0.459i)12-s + (−0.691 − 1.19i)13-s + (0.406 + 0.578i)14-s + (0.444 + 0.0532i)15-s + 0.250·16-s + (0.855 − 1.48i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.136 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.136 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.136 + 0.990i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.136 + 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.575649 - 0.501866i\)
\(L(\frac12)\) \(\approx\) \(0.575649 - 0.501866i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.681 - 1.59i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (1.52 + 2.16i)T \)
good11 \( 1 + (0.821 + 1.42i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.49 + 4.31i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.52 + 6.10i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.51 + 6.08i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.72 - 6.45i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.86 - 4.96i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 3.64T + 31T^{2} \)
37 \( 1 + (2.87 + 4.97i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.30 - 7.45i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.96 + 5.13i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 1.64T + 47T^{2} \)
53 \( 1 + (-6.46 + 11.1i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 0.782T + 59T^{2} \)
61 \( 1 - 4.76T + 61T^{2} \)
67 \( 1 + 9.95T + 67T^{2} \)
71 \( 1 + 7.26T + 71T^{2} \)
73 \( 1 + (0.392 - 0.680i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 5.86T + 79T^{2} \)
83 \( 1 + (-3.28 + 5.69i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-4.63 - 8.03i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.99 + 5.17i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.17998828720237684692985919073, −9.598555838439491370744590195802, −8.897919311528596605401579821218, −7.83169035796359497597022843704, −7.18822592962740429790235421239, −5.70674784772202814073200317814, −4.90150397616787605755114844704, −3.51840044522866111542805824152, −2.62795830361231431315216521751, −0.46979706644094715782197522417, 1.84113802483322056501018123538, 2.53031063379358800228686114144, 3.97354752387424620563300461223, 6.00295850894432333194040351282, 6.27323741990375991671281514242, 7.39657604489754115384522820432, 8.202351730248916445578980899905, 8.936570772672437184401464599398, 9.895588945331674768670482424358, 10.49898133488406021885980273331

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