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Properties

Label 2-630-21.17-c1-0-3
Degree $2$
Conductor $630$
Sign $0.958 - 0.286i$
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 + (2.55 + 1.47i)11-s + 3.93i·13-s + (1.48 + 2.19i)14-s + (−0.5 − 0.866i)16-s + (−0.199 + 0.346i)17-s + (0.0305 − 0.0176i)19-s + 0.999·20-s + 2.94·22-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 + (0.769 + 0.444i)11-s + 1.09i·13-s + (0.396 + 0.585i)14-s + (−0.125 − 0.216i)16-s + (−0.0484 + 0.0839i)17-s + (0.00700 − 0.00404i)19-s + 0.223·20-s + 0.628·22-s + (0.673 − 0.389i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 - 0.286i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.958 - 0.286i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.958 - 0.286i$
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: \((2,\ 630,\ (\ :1/2),\ 0.958 - 0.286i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.21408 + 0.323656i\)
\(L(\frac12)\) \(\approx\) \(2.21408 + 0.323656i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.55 - 1.47i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 3.93iT - 13T^{2} \)
17 \( 1 + (0.199 - 0.346i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.0305 + 0.0176i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.23 + 1.86i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 8.89iT - 29T^{2} \)
31 \( 1 + (-0.717 - 0.414i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.96 - 6.86i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 6.31T + 41T^{2} \)
43 \( 1 + 3.03T + 43T^{2} \)
47 \( 1 + (2.90 + 5.02i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.72 + 2.14i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.78 + 4.82i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (9.97 - 5.76i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.25 + 10.8i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.93iT - 71T^{2} \)
73 \( 1 + (0.297 + 0.171i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.15 + 7.19i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 10.3T + 83T^{2} \)
89 \( 1 + (3.08 + 5.34i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 15.6iT - 97T^{2} \)
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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

−10.83419335694220874666512765359, −9.670244835758686237507200037284, −9.206689595465426499697595548224, −8.019311879291515006573395985209, −6.67770301645347338877300752000, −6.21926458404540306624351601502, −5.00077234433431066982754959455, −4.08805495770776237655701686373, −2.77259223959967405240702820517, −1.77850800605494710028620234421, 1.15500138105894166125378970117, 3.05252028564252764271817927765, 4.02735348843027558156735921440, 5.04923788462329853256208738482, 5.95592483269619247131260569775, 6.97032366956590667682443769453, 7.74757411515088116382530516427, 8.721674700629626515485990172277, 9.658108481185082484376285618196, 10.76072359576067918409926481121

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