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Properties

Label 2-630-9.7-c1-0-3
Degree $2$
Conductor $630$
Sign $0.173 - 0.984i$
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 + 1.73i·3-s + (−0.499 − 0.866i)4-s + (−0.5 − 0.866i)5-s + (1.49 + 0.866i)6-s + (−0.5 + 0.866i)7-s − 0.999·8-s − 2.99·9-s − 0.999·10-s + (−1.5 + 2.59i)11-s + (1.49 − 0.866i)12-s + (3.5 + 6.06i)13-s + (0.499 + 0.866i)14-s + (1.49 − 0.866i)15-s + (−0.5 + 0.866i)16-s − 3·17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + 0.999i·3-s + (−0.249 − 0.433i)4-s + (−0.223 − 0.387i)5-s + (0.612 + 0.353i)6-s + (−0.188 + 0.327i)7-s − 0.353·8-s − 0.999·9-s − 0.316·10-s + (−0.452 + 0.783i)11-s + (0.433 − 0.249i)12-s + (0.970 + 1.68i)13-s + (0.133 + 0.231i)14-s + (0.387 − 0.223i)15-s + (−0.125 + 0.216i)16-s − 0.727·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.173 - 0.984i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (421, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.173 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.963125 + 0.808157i\)
\(L(\frac12)\) \(\approx\) \(0.963125 + 0.808157i\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.73iT \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
good11 \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.5 - 6.06i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 3T + 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 + (-3 - 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3 - 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1 + 1.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1 - 1.73i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.5 + 7.79i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 12T + 53T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7 - 12.1i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5 - 8.66i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3T + 71T^{2} \)
73 \( 1 - 11T + 73T^{2} \)
79 \( 1 + (-6.5 + 11.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-7.5 + 12.9i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 18T + 89T^{2} \)
97 \( 1 + (-3.5 + 6.06i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.02899682959039022957422339945, −9.852369508344136494152828533485, −9.207459433361387903284111137088, −8.656724238737901460993706924875, −7.19147285393687959779301365795, −5.97187231975871790783285480418, −4.97560946437625996538807041027, −4.23438658234391449612922307025, −3.30255262924032122707808302392, −1.86569059496937546483952543249, 0.61233565255700156135411541140, 2.72799219891489375905379303114, 3.57546554545646443228642307645, 5.16369681359052419291252148112, 6.10311326534115306126471650699, 6.69143538748531820733094173730, 7.967294812938959796112940887013, 8.044401932808785731129259804198, 9.278119897006643230124257513161, 10.85159304727180832664473029993

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