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Properties

Label 2-630-63.58-c1-0-26
Degree $2$
Conductor $630$
Sign $0.999 + 0.0253i$
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  + 2-s + (1.73 + 0.0100i)3-s + 4-s + (0.5 + 0.866i)5-s + (1.73 + 0.0100i)6-s + (−0.710 − 2.54i)7-s + 8-s + (2.99 + 0.0347i)9-s + (0.5 + 0.866i)10-s + (0.771 − 1.33i)11-s + (1.73 + 0.0100i)12-s + (−3.10 + 5.37i)13-s + (−0.710 − 2.54i)14-s + (0.857 + 1.50i)15-s + 16-s + (2.90 + 5.03i)17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.999 + 0.00579i)3-s + 0.5·4-s + (0.223 + 0.387i)5-s + (0.707 + 0.00409i)6-s + (−0.268 − 0.963i)7-s + 0.353·8-s + (0.999 + 0.0115i)9-s + (0.158 + 0.273i)10-s + (0.232 − 0.402i)11-s + (0.499 + 0.00289i)12-s + (−0.860 + 1.49i)13-s + (−0.189 − 0.681i)14-s + (0.221 + 0.388i)15-s + 0.250·16-s + (0.705 + 1.22i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0253i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0253i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.999 + 0.0253i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.999 + 0.0253i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.17142 - 0.0402663i\)
\(L(\frac12)\) \(\approx\) \(3.17142 - 0.0402663i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (-1.73 - 0.0100i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (0.710 + 2.54i)T \)
good11 \( 1 + (-0.771 + 1.33i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.10 - 5.37i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.90 - 5.03i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.46 + 6.00i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.62 + 6.27i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.25 + 2.17i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 8.09T + 31T^{2} \)
37 \( 1 + (4.38 - 7.58i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.70 - 2.95i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.75 - 4.76i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 5.63T + 47T^{2} \)
53 \( 1 + (2.96 + 5.13i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 8.45T + 59T^{2} \)
61 \( 1 - 4.82T + 61T^{2} \)
67 \( 1 + 2.09T + 67T^{2} \)
71 \( 1 + 7.31T + 71T^{2} \)
73 \( 1 + (1.97 + 3.41i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 2.25T + 79T^{2} \)
83 \( 1 + (4.13 + 7.16i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.13 + 3.69i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.52 - 11.3i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.50336983026971259565007702177, −9.792734053323693727059519670096, −8.919222097958355956296571989589, −7.73769413517263129277864966051, −7.00404033163966367172045180969, −6.30274016542222668905421144223, −4.71260873785383056828992576011, −3.92807959268947460283139541882, −2.97410953113611712957733774485, −1.74067567616893354586266535086, 1.80204504128574744287980142222, 2.94077237455305564198697211801, 3.75553276787293308498716782371, 5.32424887850605692203658196307, 5.61834751372428731996030268863, 7.38422503007146948070796208585, 7.68559587881508740371478890693, 8.999697970018411317854004772544, 9.653596288296880453832384572920, 10.36358881914789560964132045677

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