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Properties

Label 2-630-21.5-c1-0-1
Degree $2$
Conductor $630$
Sign $-0.807 - 0.589i$
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 + (0.499 + 0.866i)4-s + (−0.5 + 0.866i)5-s + (−2.63 − 0.189i)7-s + 0.999i·8-s + (−0.866 + 0.499i)10-s + (−4.67 + 2.69i)11-s + 2.51i·13-s + (−2.19 − 1.48i)14-s + (−0.5 + 0.866i)16-s + (2.24 + 3.89i)17-s + (−2.48 − 1.43i)19-s − 0.999·20-s − 5.39·22-s + (0.232 + 0.133i)23-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.223 + 0.387i)5-s + (−0.997 − 0.0716i)7-s + 0.353i·8-s + (−0.273 + 0.158i)10-s + (−1.40 + 0.813i)11-s + 0.698i·13-s + (−0.585 − 0.396i)14-s + (−0.125 + 0.216i)16-s + (0.545 + 0.945i)17-s + (−0.568 − 0.328i)19-s − 0.223·20-s − 1.15·22-s + (0.0483 + 0.0279i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.807 - 0.589i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.807 - 0.589i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.807 - 0.589i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (341, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.807 - 0.589i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.371917 + 1.14110i\)
\(L(\frac12)\) \(\approx\) \(0.371917 + 1.14110i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (2.63 + 0.189i)T \)
good11 \( 1 + (4.67 - 2.69i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 2.51iT - 13T^{2} \)
17 \( 1 + (-2.24 - 3.89i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.48 + 1.43i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.232 - 0.133i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 8.89iT - 29T^{2} \)
31 \( 1 + (-4.18 + 2.41i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.25 + 5.64i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 0.760T + 41T^{2} \)
43 \( 1 + 5.86T + 43T^{2} \)
47 \( 1 + (-3.99 + 6.92i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (7.27 - 4.19i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.33 - 10.9i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.27 + 1.31i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.91 + 8.50i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 4.76iT - 71T^{2} \)
73 \( 1 + (-10.0 + 5.82i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.29 - 7.44i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 9.45T + 83T^{2} \)
89 \( 1 + (3.98 - 6.90i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 6.16iT - 97T^{2} \)
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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

−10.79504446995989720272380474911, −10.27158582514865658100851163084, −9.208575232975742024456226187000, −8.092991535856707939203037410622, −7.22178371169271562327727849812, −6.52248051111738102327447023829, −5.52276920655453413492605982994, −4.43552431725754000835884834526, −3.39273352283163804108384623499, −2.31732085541320796713778986212, 0.50523534734315051909883085560, 2.64047988679926296553670527067, 3.35409253022137958313732454909, 4.68489691230345282693982248918, 5.60720194589969126518360677396, 6.37430428288614414247647036100, 7.68291239398316198509640446710, 8.403729089585077700344311778393, 9.704700118941951483230277859876, 10.21813877326757449368357205561

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