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Properties

Label 2-630-315.184-c1-0-10
Degree $2$
Conductor $630$
Sign $-0.0469 - 0.998i$
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  − i·2-s + (0.330 + 1.70i)3-s − 4-s + (0.551 + 2.16i)5-s + (1.70 − 0.330i)6-s + (−0.585 − 2.58i)7-s + i·8-s + (−2.78 + 1.12i)9-s + (2.16 − 0.551i)10-s + (−2.05 + 3.55i)11-s + (−0.330 − 1.70i)12-s + (3.36 + 1.94i)13-s + (−2.58 + 0.585i)14-s + (−3.50 + 1.65i)15-s + 16-s + (2.40 − 1.38i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.190 + 0.981i)3-s − 0.5·4-s + (0.246 + 0.969i)5-s + (0.694 − 0.134i)6-s + (−0.221 − 0.975i)7-s + 0.353i·8-s + (−0.927 + 0.374i)9-s + (0.685 − 0.174i)10-s + (−0.618 + 1.07i)11-s + (−0.0953 − 0.490i)12-s + (0.932 + 0.538i)13-s + (−0.689 + 0.156i)14-s + (−0.904 + 0.426i)15-s + 0.250·16-s + (0.583 − 0.336i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0469 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0469 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.0469 - 0.998i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (499, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.0469 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.818121 + 0.857437i\)
\(L(\frac12)\) \(\approx\) \(0.818121 + 0.857437i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + (-0.330 - 1.70i)T \)
5 \( 1 + (-0.551 - 2.16i)T \)
7 \( 1 + (0.585 + 2.58i)T \)
good11 \( 1 + (2.05 - 3.55i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.36 - 1.94i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.40 + 1.38i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.40 - 2.44i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (5.13 - 2.96i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.36 - 4.08i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 3.89T + 31T^{2} \)
37 \( 1 + (0.356 + 0.206i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.43 - 5.94i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.08 - 0.627i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 5.10iT - 47T^{2} \)
53 \( 1 + (-4.20 + 2.42i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 12.4T + 59T^{2} \)
61 \( 1 + 1.08T + 61T^{2} \)
67 \( 1 + 12.5iT - 67T^{2} \)
71 \( 1 - 16.3T + 71T^{2} \)
73 \( 1 + (-11.1 + 6.41i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 6.02T + 79T^{2} \)
83 \( 1 + (13.4 - 7.74i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-3.68 + 6.38i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.62 - 2.09i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.75590330118098058399211772499, −9.866008752295557152365238130537, −9.737904929674154164447453348654, −8.313595473672461435271212930131, −7.39760245907770093956463089984, −6.25348540630575958103817794966, −5.05044635138458933566005400490, −3.93773975057150969698024714017, −3.32977354928246800668354054205, −1.97999827982950704562228088272, 0.62633834443251700961228327028, 2.28832299473395012217259436838, 3.68524193121364236115277016368, 5.43970458464462058624302704470, 5.74949654790507591018689667154, 6.64818848951641885099950699128, 8.110040344765689196896641507142, 8.361708999757779922424647892725, 9.017181220758507763941365265229, 10.20490493632959605614981760960

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