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Properties

Label 2-630-315.299-c1-0-8
Degree $2$
Conductor $630$
Sign $-0.962 + 0.269i$
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 + (−0.500 + 1.65i)3-s + (−0.499 + 0.866i)4-s + (−1.26 + 1.84i)5-s + (−1.68 + 0.395i)6-s + (2.59 − 0.497i)7-s − 0.999·8-s + (−2.49 − 1.66i)9-s + (−2.22 − 0.167i)10-s + 6.34i·11-s + (−1.18 − 1.26i)12-s + (0.442 + 0.766i)13-s + (1.72 + 2.00i)14-s + (−2.43 − 3.01i)15-s + (−0.5 − 0.866i)16-s + (1.83 − 1.06i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.289 + 0.957i)3-s + (−0.249 + 0.433i)4-s + (−0.563 + 0.826i)5-s + (−0.688 + 0.161i)6-s + (0.982 − 0.187i)7-s − 0.353·8-s + (−0.832 − 0.553i)9-s + (−0.705 − 0.0530i)10-s + 1.91i·11-s + (−0.342 − 0.364i)12-s + (0.122 + 0.212i)13-s + (0.462 + 0.535i)14-s + (−0.627 − 0.778i)15-s + (−0.125 − 0.216i)16-s + (0.446 − 0.257i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.962 + 0.269i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.962 + 0.269i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.962 + 0.269i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.962 + 0.269i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.169390 - 1.23257i\)
\(L(\frac12)\) \(\approx\) \(0.169390 - 1.23257i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.500 - 1.65i)T \)
5 \( 1 + (1.26 - 1.84i)T \)
7 \( 1 + (-2.59 + 0.497i)T \)
good11 \( 1 - 6.34iT - 11T^{2} \)
13 \( 1 + (-0.442 - 0.766i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.83 + 1.06i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.84 + 2.21i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 5.06T + 23T^{2} \)
29 \( 1 + (-6.27 - 3.62i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.47 + 2.58i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.490 - 0.283i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.01 + 6.94i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-8.74 - 5.04i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.99 + 2.88i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.642 - 1.11i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.08 - 5.34i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.40 - 4.27i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.85 + 1.06i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 9.05iT - 71T^{2} \)
73 \( 1 + (-3.15 - 5.45i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.75 - 13.4i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.79 - 2.76i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (3.80 - 6.59i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.32 - 4.01i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.92653252959823939672591132548, −10.33729969759946823225313629987, −9.378147876756682254105988843221, −8.303871505422951993260949573922, −7.41546325696642978513984214444, −6.68867448773253274456894024710, −5.46336155961356952318109907422, −4.43326761866586364061103275617, −4.04196390931071563775536942442, −2.44788770398465949069856928909, 0.64123408148146090110342354045, 1.80414182346773011154079066535, 3.30217466233867089545098556500, 4.51278126079340878821431503903, 5.61344625267918705548572422915, 6.15488850289697097298523485956, 7.85160816323233815999432177748, 8.251271406802700656897897387998, 8.975083712955289231389066006196, 10.63391779298830929914419404392

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