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Properties

Label 2-630-63.59-c1-0-5
Degree $2$
Conductor $630$
Sign $-0.992 - 0.120i$
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  + (0.866 + 0.5i)2-s + (0.0350 + 1.73i)3-s + (0.499 + 0.866i)4-s − 5-s + (−0.835 + 1.51i)6-s + (−1.17 + 2.36i)7-s + 0.999i·8-s + (−2.99 + 0.121i)9-s + (−0.866 − 0.5i)10-s + 0.751i·11-s + (−1.48 + 0.896i)12-s + (−2.72 − 1.57i)13-s + (−2.20 + 1.46i)14-s + (−0.0350 − 1.73i)15-s + (−0.5 + 0.866i)16-s + (0.433 − 0.750i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.0202 + 0.999i)3-s + (0.249 + 0.433i)4-s − 0.447·5-s + (−0.341 + 0.619i)6-s + (−0.445 + 0.895i)7-s + 0.353i·8-s + (−0.999 + 0.0405i)9-s + (−0.273 − 0.158i)10-s + 0.226i·11-s + (−0.427 + 0.258i)12-s + (−0.754 − 0.435i)13-s + (−0.589 + 0.390i)14-s + (−0.00906 − 0.447i)15-s + (−0.125 + 0.216i)16-s + (0.105 − 0.181i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.120i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.992 - 0.120i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.992 - 0.120i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (311, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.992 - 0.120i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0829468 + 1.37549i\)
\(L(\frac12)\) \(\approx\) \(0.0829468 + 1.37549i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (-0.0350 - 1.73i)T \)
5 \( 1 + T \)
7 \( 1 + (1.17 - 2.36i)T \)
good11 \( 1 - 0.751iT - 11T^{2} \)
13 \( 1 + (2.72 + 1.57i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.433 + 0.750i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.23 + 0.713i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 5.06iT - 23T^{2} \)
29 \( 1 + (5.23 - 3.02i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-5.38 + 3.10i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.60 - 6.24i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.08 - 3.61i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.69 - 6.40i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.72 - 2.99i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.790 + 0.456i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.21 + 7.30i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-9.30 - 5.37i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.44 - 9.43i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 5.43iT - 71T^{2} \)
73 \( 1 + (-0.539 - 0.311i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.00292 - 0.00506i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.01 - 1.76i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.22 + 2.12i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-9.23 + 5.33i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.26466234990079148190514131557, −9.940325935866566029716048381443, −9.429406356523172312137676351694, −8.360943609958829146158858183927, −7.52431015587969188846468911588, −6.28488968401593176356750955139, −5.37760149347131624718492966658, −4.66283848662271530340253862428, −3.47672851252750265385211160749, −2.66061217056700177760987618225, 0.59348831546055381236240889898, 2.19886753705151887303471928536, 3.39495384802926482232704978593, 4.41929870226896205818744535194, 5.67368863422241150781556956614, 6.69993268989023967679897939235, 7.29974578758001606638381047083, 8.220731928682377653537497949388, 9.382028569881261786565349890657, 10.42763928647418035184895388674

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