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Properties

Label 2-630-315.209-c1-0-11
Degree $2$
Conductor $630$
Sign $-0.463 - 0.886i$
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 + (1.24 + 1.20i)3-s + (−0.499 − 0.866i)4-s + (−0.984 − 2.00i)5-s + (−1.66 + 0.477i)6-s + (0.433 + 2.60i)7-s + 0.999·8-s + (0.106 + 2.99i)9-s + (2.23 + 0.150i)10-s + (0.566 + 0.326i)11-s + (0.418 − 1.68i)12-s + (1.74 + 3.02i)13-s + (−2.47 − 0.929i)14-s + (1.18 − 3.68i)15-s + (−0.5 + 0.866i)16-s − 1.32i·17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.719 + 0.694i)3-s + (−0.249 − 0.433i)4-s + (−0.440 − 0.897i)5-s + (−0.679 + 0.195i)6-s + (0.163 + 0.986i)7-s + 0.353·8-s + (0.0354 + 0.999i)9-s + (0.705 + 0.0477i)10-s + (0.170 + 0.0985i)11-s + (0.120 − 0.485i)12-s + (0.484 + 0.838i)13-s + (−0.662 − 0.248i)14-s + (0.306 − 0.951i)15-s + (−0.125 + 0.216i)16-s − 0.321i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.463 - 0.886i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.463 - 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.463 - 0.886i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.463 - 0.886i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.711766 + 1.17516i\)
\(L(\frac12)\) \(\approx\) \(0.711766 + 1.17516i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.24 - 1.20i)T \)
5 \( 1 + (0.984 + 2.00i)T \)
7 \( 1 + (-0.433 - 2.60i)T \)
good11 \( 1 + (-0.566 - 0.326i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.74 - 3.02i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 1.32iT - 17T^{2} \)
19 \( 1 + 0.281iT - 19T^{2} \)
23 \( 1 + (-0.672 - 1.16i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-7.41 - 4.28i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (6.47 - 3.73i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 4.13iT - 37T^{2} \)
41 \( 1 + (-1.75 - 3.03i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.75 - 3.32i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (6.86 + 3.96i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 6.01T + 53T^{2} \)
59 \( 1 + (6.08 + 10.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.98 + 2.29i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.66 - 3.27i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 10.9iT - 71T^{2} \)
73 \( 1 - 16.8T + 73T^{2} \)
79 \( 1 + (-7.26 + 12.5i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.51 + 0.876i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 11.5T + 89T^{2} \)
97 \( 1 + (-2.59 + 4.49i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.74674556115456724556056666899, −9.505432196081796990005040742524, −9.061852078702847767063155571563, −8.459132225895564833520258537338, −7.67763622649838079200761042301, −6.43775059979597124234000840641, −5.17308952813253320714237810545, −4.62137974934574918282683710613, −3.34785427846325466323224391224, −1.72937586881291995686949383670, 0.828790613969269865019528383523, 2.37154759113460412630559994536, 3.43734981263099569439369129028, 4.14466444812150996599983515044, 6.06954738210196990554077153976, 7.08044929822109149525710156322, 7.75953341406563059321465863054, 8.401389847807167228215503194328, 9.509855404025074637221666384006, 10.48309457473599021955422835507

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