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Properties

Label 2-630-21.17-c1-0-4
Degree $2$
Conductor $630$
Sign $0.589 + 0.807i$
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  + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (0.5 + 0.866i)5-s + (−0.189 − 2.63i)7-s − 0.999i·8-s + (0.866 + 0.499i)10-s + (3.44 + 1.99i)11-s + 0.0681i·13-s + (−1.48 − 2.19i)14-s + (−0.5 − 0.866i)16-s + (3.66 − 6.34i)17-s + (−1.76 + 1.01i)19-s + 0.999·20-s + 3.98·22-s + (3.23 − 1.86i)23-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (0.223 + 0.387i)5-s + (−0.0716 − 0.997i)7-s − 0.353i·8-s + (0.273 + 0.158i)10-s + (1.03 + 0.600i)11-s + 0.0189i·13-s + (−0.396 − 0.585i)14-s + (−0.125 − 0.216i)16-s + (0.888 − 1.53i)17-s + (−0.404 + 0.233i)19-s + 0.223·20-s + 0.848·22-s + (0.673 − 0.389i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.589 + 0.807i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.589 + 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.589 + 0.807i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.589 + 0.807i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.01723 - 1.02551i\)
\(L(\frac12)\) \(\approx\) \(2.01723 - 1.02551i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (0.189 + 2.63i)T \)
good11 \( 1 + (-3.44 - 1.99i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 0.0681iT - 13T^{2} \)
17 \( 1 + (-3.66 + 6.34i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.76 - 1.01i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.23 + 1.86i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 0.898iT - 29T^{2} \)
31 \( 1 + (4.18 + 2.41i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.03 - 3.52i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 1.68T + 41T^{2} \)
43 \( 1 + 0.964T + 43T^{2} \)
47 \( 1 + (0.830 + 1.43i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-11.4 - 6.61i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.32 - 9.21i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.51 + 3.76i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.33 - 9.23i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 9.93iT - 71T^{2} \)
73 \( 1 + (10.0 + 5.82i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (8.77 + 15.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 14.3T + 83T^{2} \)
89 \( 1 + (0.913 + 1.58i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 17.1iT - 97T^{2} \)
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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

−10.45429035892806582967049184161, −9.847673819477464924134984305639, −8.970006698981640671222411124409, −7.39993505388626807412361175805, −6.98246633133563074054276402263, −5.89146831497040971342269870383, −4.71645937743421513305558191790, −3.85775871921442236652622319412, −2.75213931208335548540094311964, −1.20869466097397879082525053383, 1.69466258838330040570769470221, 3.19238685554930521490096886892, 4.18815409113366436890758604125, 5.49667318692293661666002901186, 5.96760211274056451832073916506, 6.96124951379636108938889697386, 8.258549450091270434563199503383, 8.798872954116694337390979039500, 9.709104600585880493096267778239, 10.93105166081413893894690869072

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