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Properties

Label 2-630-5.3-c2-0-16
Degree $2$
Conductor $630$
Sign $0.751 + 0.659i$
Analytic cond. $17.1662$
Root an. cond. $4.14321$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 − i)2-s − 2i·4-s + (4.93 + 0.829i)5-s + (1.87 − 1.87i)7-s + (−2 − 2i)8-s + (5.76 − 4.10i)10-s + 5.74·11-s + (15.0 + 15.0i)13-s − 3.74i·14-s − 4·16-s + (4.78 − 4.78i)17-s + 17.4i·19-s + (1.65 − 9.86i)20-s + (5.74 − 5.74i)22-s + (−13.2 − 13.2i)23-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s − 0.5i·4-s + (0.986 + 0.165i)5-s + (0.267 − 0.267i)7-s + (−0.250 − 0.250i)8-s + (0.576 − 0.410i)10-s + 0.522·11-s + (1.15 + 1.15i)13-s − 0.267i·14-s − 0.250·16-s + (0.281 − 0.281i)17-s + 0.915i·19-s + (0.0829 − 0.493i)20-s + (0.261 − 0.261i)22-s + (−0.575 − 0.575i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.751 + 0.659i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.751 + 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.751 + 0.659i$
Analytic conductor: \(17.1662\)
Root analytic conductor: \(4.14321\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1),\ 0.751 + 0.659i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.129067532\)
\(L(\frac12)\) \(\approx\) \(3.129067532\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 + i)T \)
3 \( 1 \)
5 \( 1 + (-4.93 - 0.829i)T \)
7 \( 1 + (-1.87 + 1.87i)T \)
good11 \( 1 - 5.74T + 121T^{2} \)
13 \( 1 + (-15.0 - 15.0i)T + 169iT^{2} \)
17 \( 1 + (-4.78 + 4.78i)T - 289iT^{2} \)
19 \( 1 - 17.4iT - 361T^{2} \)
23 \( 1 + (13.2 + 13.2i)T + 529iT^{2} \)
29 \( 1 + 37.7iT - 841T^{2} \)
31 \( 1 + 27.0T + 961T^{2} \)
37 \( 1 + (-11.3 + 11.3i)T - 1.36e3iT^{2} \)
41 \( 1 - 53.0T + 1.68e3T^{2} \)
43 \( 1 + (-37.1 - 37.1i)T + 1.84e3iT^{2} \)
47 \( 1 + (-9.39 + 9.39i)T - 2.20e3iT^{2} \)
53 \( 1 + (43.8 + 43.8i)T + 2.80e3iT^{2} \)
59 \( 1 - 62.9iT - 3.48e3T^{2} \)
61 \( 1 + 1.67T + 3.72e3T^{2} \)
67 \( 1 + (-28.4 + 28.4i)T - 4.48e3iT^{2} \)
71 \( 1 + 47.2T + 5.04e3T^{2} \)
73 \( 1 + (31.2 + 31.2i)T + 5.32e3iT^{2} \)
79 \( 1 + 107. iT - 6.24e3T^{2} \)
83 \( 1 + (18.2 + 18.2i)T + 6.88e3iT^{2} \)
89 \( 1 + 174. iT - 7.92e3T^{2} \)
97 \( 1 + (91.5 - 91.5i)T - 9.40e3iT^{2} \)
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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

−10.38255997105140796514641060806, −9.535333259725006627333767908498, −8.846097308676951710923043260799, −7.56057207394070315141568544823, −6.27799615404349271151456868681, −5.93182311378356928547815544410, −4.53161692740909977597552956611, −3.70217678834252557003393070493, −2.25189424172707201455983494374, −1.28385085787346834905821848483, 1.31644363451498771008277032668, 2.79833974829653495652066643543, 3.97142368469780086249240030874, 5.29745876029920663375236354260, 5.80358449675821514073349904532, 6.71497270139679648375629511611, 7.82392943383358104141837835376, 8.762239130396079586380204649935, 9.412270931978661697178389496675, 10.60963948618594280377360166928

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