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Properties

Label 2-630-315.202-c1-0-40
Degree $2$
Conductor $630$
Sign $-0.969 + 0.244i$
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  + (−0.258 + 0.965i)2-s + (0.0305 − 1.73i)3-s + (−0.866 − 0.499i)4-s + (−1.58 + 1.57i)5-s + (1.66 + 0.477i)6-s + (1.12 − 2.39i)7-s + (0.707 − 0.707i)8-s + (−2.99 − 0.105i)9-s + (−1.10 − 1.94i)10-s + (−0.243 − 0.421i)11-s + (−0.892 + 1.48i)12-s + (−1.17 + 0.314i)13-s + (2.01 + 1.70i)14-s + (2.67 + 2.80i)15-s + (0.500 + 0.866i)16-s + (−2.59 + 2.59i)17-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (0.0176 − 0.999i)3-s + (−0.433 − 0.249i)4-s + (−0.710 + 0.703i)5-s + (0.679 + 0.195i)6-s + (0.426 − 0.904i)7-s + (0.249 − 0.249i)8-s + (−0.999 − 0.0352i)9-s + (−0.350 − 0.614i)10-s + (−0.0734 − 0.127i)11-s + (−0.257 + 0.428i)12-s + (−0.326 + 0.0873i)13-s + (0.539 + 0.456i)14-s + (0.690 + 0.723i)15-s + (0.125 + 0.216i)16-s + (−0.628 + 0.628i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 + 0.244i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.969 + 0.244i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.969 + 0.244i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (517, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.969 + 0.244i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0159749 - 0.128833i\)
\(L(\frac12)\) \(\approx\) \(0.0159749 - 0.128833i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 - 0.965i)T \)
3 \( 1 + (-0.0305 + 1.73i)T \)
5 \( 1 + (1.58 - 1.57i)T \)
7 \( 1 + (-1.12 + 2.39i)T \)
good11 \( 1 + (0.243 + 0.421i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.17 - 0.314i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (2.59 - 2.59i)T - 17iT^{2} \)
19 \( 1 + 7.77T + 19T^{2} \)
23 \( 1 + (-1.10 - 4.12i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (-0.0975 + 0.0563i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.577 - 0.333i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (6.32 + 6.32i)T + 37iT^{2} \)
41 \( 1 + (8.77 + 5.06i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.51 - 0.942i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (3.43 + 0.919i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (6.56 - 6.56i)T - 53iT^{2} \)
59 \( 1 + (-0.680 + 1.17i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.58 - 3.22i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (13.3 - 3.57i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 4.92T + 71T^{2} \)
73 \( 1 + (-3.32 - 3.32i)T + 73iT^{2} \)
79 \( 1 + (-9.28 + 5.35i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.950 + 3.54i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 - 2.14T + 89T^{2} \)
97 \( 1 + (-13.5 - 3.62i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.48321066401706250522366210783, −8.918667411672064852652645851238, −8.186063303365647257287969152377, −7.42444897684766482584412335580, −6.84788145891403366700182571406, −6.04520635254595173960309823760, −4.63143635800299695811168119340, −3.55199253674369631156191939973, −1.92042898538905337391263993597, −0.07136966635428001891264699739, 2.17872895559872190133995776708, 3.38140097296835911186907770854, 4.75858673449732805555274197398, 4.81413079458498619700203790251, 6.38489238808555479438726461925, 8.014375219032476897255800923884, 8.649464604135943135151740972294, 9.156999777957460462929381102444, 10.20179175117742736593646519733, 11.01278710616306088910774854694

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