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Properties

Label 2-630-63.16-c1-0-26
Degree $2$
Conductor $630$
Sign $-0.998 + 0.0617i$
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  + (−0.5 − 0.866i)2-s + (−0.570 − 1.63i)3-s + (−0.499 + 0.866i)4-s − 5-s + (−1.13 + 1.31i)6-s + (2.25 − 1.38i)7-s + 0.999·8-s + (−2.34 + 1.86i)9-s + (0.5 + 0.866i)10-s + 5.42·11-s + (1.70 + 0.323i)12-s + (−2.92 − 5.06i)13-s + (−2.32 − 1.26i)14-s + (0.570 + 1.63i)15-s + (−0.5 − 0.866i)16-s + (−1.75 − 3.03i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.329 − 0.944i)3-s + (−0.249 + 0.433i)4-s − 0.447·5-s + (−0.461 + 0.535i)6-s + (0.853 − 0.521i)7-s + 0.353·8-s + (−0.782 + 0.622i)9-s + (0.158 + 0.273i)10-s + 1.63·11-s + (0.491 + 0.0933i)12-s + (−0.811 − 1.40i)13-s + (−0.621 − 0.338i)14-s + (0.147 + 0.422i)15-s + (−0.125 − 0.216i)16-s + (−0.425 − 0.736i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0617i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0617i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.998 + 0.0617i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (331, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.998 + 0.0617i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0266358 - 0.861840i\)
\(L(\frac12)\) \(\approx\) \(0.0266358 - 0.861840i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + (0.570 + 1.63i)T \)
5 \( 1 + T \)
7 \( 1 + (-2.25 + 1.38i)T \)
good11 \( 1 - 5.42T + 11T^{2} \)
13 \( 1 + (2.92 + 5.06i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.75 + 3.03i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.62 + 4.54i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 7.93T + 23T^{2} \)
29 \( 1 + (4.29 - 7.44i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.79 - 3.11i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.172 + 0.297i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.330 - 0.571i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.57 + 6.18i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.973 + 1.68i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.405 - 0.702i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.0329 + 0.0570i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.11 + 3.66i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.68 - 2.92i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 5.85T + 71T^{2} \)
73 \( 1 + (-1.16 - 2.01i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (8.48 + 14.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.66 - 8.08i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-7.34 + 12.7i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.116 + 0.201i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.41540432995578884570844826551, −9.226390166356981631595586457603, −8.387627639617028997182078961414, −7.42999434265304556260750085898, −7.02268624077780247298869494337, −5.53601357851999188987469333373, −4.50240603928340622026940991211, −3.20929852593485421896693310979, −1.79973761486072906112757011926, −0.57161665237241822050600915052, 1.83564850306073028234762608199, 4.06507221470593947215057963618, 4.32208200425616308402086486216, 5.73615171877442460080243329924, 6.37257547253789021653837042029, 7.62825398219148299453764288601, 8.494272790293790469899721653702, 9.364613986321343314077217045318, 9.810696917747779629672434492122, 11.09679467601640865522638211426

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