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Properties

Label 2-630-105.59-c1-0-13
Degree $2$
Conductor $630$
Sign $-0.989 - 0.145i$
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 + (−0.499 + 0.866i)4-s + (−2.11 + 0.728i)5-s + (2.30 + 1.29i)7-s + 0.999·8-s + (1.68 + 1.46i)10-s + (−1.11 − 0.641i)11-s − 6.14·13-s + (−0.0298 − 2.64i)14-s + (−0.5 − 0.866i)16-s + (−5.64 − 3.26i)17-s + (5.22 − 3.01i)19-s + (0.426 − 2.19i)20-s + 1.28i·22-s + (−1.43 − 2.49i)23-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.945 + 0.325i)5-s + (0.871 + 0.490i)7-s + 0.353·8-s + (0.533 + 0.463i)10-s + (−0.335 − 0.193i)11-s − 1.70·13-s + (−0.00798 − 0.707i)14-s + (−0.125 − 0.216i)16-s + (−1.37 − 0.790i)17-s + (1.19 − 0.692i)19-s + (0.0953 − 0.490i)20-s + 0.273i·22-s + (−0.300 − 0.519i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 - 0.145i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.989 - 0.145i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.989 - 0.145i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.989 - 0.145i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0107966 + 0.147823i\)
\(L(\frac12)\) \(\approx\) \(0.0107966 + 0.147823i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + (2.11 - 0.728i)T \)
7 \( 1 + (-2.30 - 1.29i)T \)
good11 \( 1 + (1.11 + 0.641i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 6.14T + 13T^{2} \)
17 \( 1 + (5.64 + 3.26i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.22 + 3.01i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.43 + 2.49i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 1.35iT - 29T^{2} \)
31 \( 1 + (7.49 + 4.32i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (8.25 - 4.76i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 8.71T + 41T^{2} \)
43 \( 1 - 5.35iT - 43T^{2} \)
47 \( 1 + (-0.698 + 0.403i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.33 + 5.77i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.798 + 1.38i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.50 - 3.17i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.67 + 2.69i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 15.6iT - 71T^{2} \)
73 \( 1 + (-6.20 + 10.7i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.59 + 9.68i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 3.74iT - 83T^{2} \)
89 \( 1 + (1.81 + 3.15i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 8.76T + 97T^{2} \)
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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

−10.24772171739968851950161549391, −9.251830693646353498875810189961, −8.479126864567236072638415411467, −7.53658159968241317524131135688, −6.98176254042462511620826684536, −5.14315866038981227016440315409, −4.57566630356025004012131022907, −3.11807446716457654465666545318, −2.18141177492651103074309472833, −0.087040441079224513116139766274, 1.81980726278983370037476348589, 3.73013148414163065548707968195, 4.76605758041849071943036791929, 5.41655469794303777669889131944, 7.11507418547415407229349926450, 7.41667173686515700611344686714, 8.294215508703426495128393178574, 9.104355158572010231763032210159, 10.18947589977345146509737735398, 10.91472803174505847830758233961

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