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Properties

Label 2-630-9.7-c1-0-5
Degree $2$
Conductor $630$
Sign $-0.195 - 0.980i$
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 + (−1.70 + 0.319i)3-s + (−0.499 − 0.866i)4-s + (0.5 + 0.866i)5-s + (0.574 − 1.63i)6-s + (0.5 − 0.866i)7-s + 0.999·8-s + (2.79 − 1.08i)9-s − 0.999·10-s + (−1.83 + 3.16i)11-s + (1.12 + 1.31i)12-s + (−1.78 − 3.09i)13-s + (0.499 + 0.866i)14-s + (−1.12 − 1.31i)15-s + (−0.5 + 0.866i)16-s + 4.46·17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.982 + 0.184i)3-s + (−0.249 − 0.433i)4-s + (0.223 + 0.387i)5-s + (0.234 − 0.667i)6-s + (0.188 − 0.327i)7-s + 0.353·8-s + (0.932 − 0.362i)9-s − 0.316·10-s + (−0.551 + 0.955i)11-s + (0.325 + 0.379i)12-s + (−0.495 − 0.858i)13-s + (0.133 + 0.231i)14-s + (−0.291 − 0.339i)15-s + (−0.125 + 0.216i)16-s + 1.08·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.195 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.195 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.195 - 0.980i$
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: \((2,\ 630,\ (\ :1/2),\ -0.195 - 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.531455 + 0.647571i\)
\(L(\frac12)\) \(\approx\) \(0.531455 + 0.647571i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.70 - 0.319i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
good11 \( 1 + (1.83 - 3.16i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.78 + 3.09i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 4.46T + 17T^{2} \)
19 \( 1 - 5.59T + 19T^{2} \)
23 \( 1 + (-1.33 - 2.30i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.223 + 0.387i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.59 - 7.95i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 10.1T + 37T^{2} \)
41 \( 1 + (-2.28 - 3.96i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.61 - 6.26i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (6.13 - 10.6i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 4.98T + 53T^{2} \)
59 \( 1 + (-6.45 - 11.1i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.596 - 1.03i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.88 + 8.45i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.89T + 71T^{2} \)
73 \( 1 - 6.14T + 73T^{2} \)
79 \( 1 + (-1.11 + 1.93i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.68 + 9.85i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 13.3T + 89T^{2} \)
97 \( 1 + (-4.26 + 7.38i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.38455694537779658148862139891, −10.22019199777179600659257029067, −9.349875696664518347117340965336, −7.79916300288835783067949905848, −7.36896422955350332288541820835, −6.40744752634161643339440376987, −5.29649853529662745371680288857, −4.88156226036160984150746461965, −3.23481107084998574504496773767, −1.24821117042180304885928083041, 0.68924996627948437687227536482, 2.09535438998476674957907092221, 3.60899904683549633348664958428, 5.02340678729369820904602830863, 5.54838415999037613265362174987, 6.79608922507215770609237930842, 7.79421524255102022097789808251, 8.700359926407080127470907021249, 9.758801941884989306909771877978, 10.31696712851942367086398007120

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