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Properties

Label 2-630-9.4-c1-0-9
Degree $2$
Conductor $630$
Sign $0.600 + 0.799i$
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 + (−0.403 − 1.68i)3-s + (−0.499 + 0.866i)4-s + (−0.5 + 0.866i)5-s + (−1.25 + 1.19i)6-s + (0.5 + 0.866i)7-s + 0.999·8-s + (−2.67 + 1.35i)9-s + 0.999·10-s + (2.25 + 3.90i)11-s + (1.66 + 0.492i)12-s + (1.56 − 2.70i)13-s + (0.499 − 0.866i)14-s + (1.66 + 0.492i)15-s + (−0.5 − 0.866i)16-s + 5.34·17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.232 − 0.972i)3-s + (−0.249 + 0.433i)4-s + (−0.223 + 0.387i)5-s + (−0.513 + 0.486i)6-s + (0.188 + 0.327i)7-s + 0.353·8-s + (−0.891 + 0.452i)9-s + 0.316·10-s + (0.680 + 1.17i)11-s + (0.479 + 0.142i)12-s + (0.433 − 0.751i)13-s + (0.133 − 0.231i)14-s + (0.428 + 0.127i)15-s + (−0.125 − 0.216i)16-s + 1.29·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.600 + 0.799i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.600 + 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.600 + 0.799i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.600 + 0.799i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01545 - 0.507012i\)
\(L(\frac12)\) \(\approx\) \(1.01545 - 0.507012i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.403 + 1.68i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
good11 \( 1 + (-2.25 - 3.90i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.56 + 2.70i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 5.34T + 17T^{2} \)
19 \( 1 - 0.320T + 19T^{2} \)
23 \( 1 + (-0.950 + 1.64i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.24 - 3.88i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.51 + 6.08i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 1.80T + 37T^{2} \)
41 \( 1 + (2.57 - 4.46i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.33 + 4.04i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.02 - 6.97i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 8.24T + 53T^{2} \)
59 \( 1 + (-6.38 + 11.0i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.09 - 1.89i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.48 + 6.02i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 12.0T + 71T^{2} \)
73 \( 1 - 2.02T + 73T^{2} \)
79 \( 1 + (0.707 + 1.22i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.80 - 3.12i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 2T + 89T^{2} \)
97 \( 1 + (-9.17 - 15.8i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.53155235960581050209775205116, −9.742513379804297856895197686799, −8.617293533400651241584719511648, −7.83924323896062867216585904093, −7.09732248562106322994903440788, −6.08694312973204349693636028452, −4.94144265516160354486873159123, −3.49314774233718518484917965238, −2.35700759333349829864293997825, −1.11043226246271327986541454937, 0.989614684901411052374532820081, 3.38920084612496128407319474944, 4.24672580936921697295887760453, 5.34378414272586180226965310804, 6.08956259117360666084727359451, 7.17184341186473887250959590018, 8.483734759869625531084954298610, 8.759010780654610133699139923418, 9.838363350542439772189350559272, 10.50850403494860564981074668811

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