Properties

Label 2-630-315.212-c1-0-38
Degree $2$
Conductor $630$
Sign $0.647 + 0.762i$
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  + (0.707 − 0.707i)2-s + (1.57 + 0.717i)3-s − 1.00i·4-s + (1.60 − 1.55i)5-s + (1.62 − 0.607i)6-s + (2.55 − 0.673i)7-s + (−0.707 − 0.707i)8-s + (1.97 + 2.26i)9-s + (0.0316 − 2.23i)10-s + (−4.13 − 2.38i)11-s + (0.717 − 1.57i)12-s + (0.348 + 1.30i)13-s + (1.33 − 2.28i)14-s + (3.64 − 1.30i)15-s − 1.00·16-s + (−4.94 − 1.32i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (0.910 + 0.414i)3-s − 0.500i·4-s + (0.717 − 0.697i)5-s + (0.662 − 0.247i)6-s + (0.967 − 0.254i)7-s + (−0.250 − 0.250i)8-s + (0.656 + 0.754i)9-s + (0.00999 − 0.707i)10-s + (−1.24 − 0.719i)11-s + (0.207 − 0.455i)12-s + (0.0967 + 0.361i)13-s + (0.356 − 0.610i)14-s + (0.941 − 0.337i)15-s − 0.250·16-s + (−1.19 − 0.321i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.647 + 0.762i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.647 + 0.762i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.647 + 0.762i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (527, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.647 + 0.762i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.65999 - 1.23047i\)
\(L(\frac12)\) \(\approx\) \(2.65999 - 1.23047i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 0.707i)T \)
3 \( 1 + (-1.57 - 0.717i)T \)
5 \( 1 + (-1.60 + 1.55i)T \)
7 \( 1 + (-2.55 + 0.673i)T \)
good11 \( 1 + (4.13 + 2.38i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.348 - 1.30i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (4.94 + 1.32i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-1.62 - 0.939i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.33 - 4.99i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-4.90 - 8.48i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 2.30T + 31T^{2} \)
37 \( 1 + (3.39 - 0.910i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (4.61 + 2.66i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.375 - 0.100i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (-1.56 + 1.56i)T - 47iT^{2} \)
53 \( 1 + (0.0260 - 0.0971i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + 1.49T + 59T^{2} \)
61 \( 1 - 13.4T + 61T^{2} \)
67 \( 1 + (-2.59 - 2.59i)T + 67iT^{2} \)
71 \( 1 + 14.1iT - 71T^{2} \)
73 \( 1 + (-5.55 - 1.48i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 - 3.85iT - 79T^{2} \)
83 \( 1 + (1.17 - 4.36i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (7.07 - 12.2i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.94 - 10.9i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.55732519275097859982543239947, −9.646270936234394846318765742611, −8.752979775905528961165517243462, −8.178987606867533326030598752943, −6.96839666917687323453075059697, −5.35263473741826383351267749379, −4.97749615258468282391483141902, −3.82554940789560133690584501003, −2.56860832656798750042763843075, −1.55759711659664769476965847613, 2.11810738104152055881061037220, 2.70251167968567720241927182405, 4.23280145099971946106738449636, 5.23622799217213361275424539263, 6.35665224831804858794222079412, 7.18069297001576650962879147546, 8.023723957018278209438240093380, 8.654926601699968936813039674983, 9.840810488293587032097971576627, 10.60918753652406698020326167807

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