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Properties

Label 2-630-35.17-c1-0-19
Degree $2$
Conductor $630$
Sign $-0.963 + 0.266i$
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  + (0.258 − 0.965i)2-s + (−0.866 − 0.499i)4-s + (−0.264 − 2.22i)5-s + (0.698 − 2.55i)7-s + (−0.707 + 0.707i)8-s + (−2.21 − 0.318i)10-s + (0.371 − 0.643i)11-s + (−2.05 − 2.05i)13-s + (−2.28 − 1.33i)14-s + (0.500 + 0.866i)16-s + (1.69 + 6.33i)17-s + (−0.946 − 1.63i)19-s + (−0.880 + 2.05i)20-s + (−0.525 − 0.525i)22-s + (−5.11 − 1.36i)23-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + (−0.433 − 0.249i)4-s + (−0.118 − 0.992i)5-s + (0.264 − 0.964i)7-s + (−0.249 + 0.249i)8-s + (−0.699 − 0.100i)10-s + (0.112 − 0.194i)11-s + (−0.570 − 0.570i)13-s + (−0.610 − 0.356i)14-s + (0.125 + 0.216i)16-s + (0.411 + 1.53i)17-s + (−0.217 − 0.375i)19-s + (−0.196 + 0.459i)20-s + (−0.112 − 0.112i)22-s + (−1.06 − 0.285i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.963 + 0.266i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.963 + 0.266i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.963 + 0.266i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.963 + 0.266i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.167173 - 1.23105i\)
\(L(\frac12)\) \(\approx\) \(0.167173 - 1.23105i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 + 0.965i)T \)
3 \( 1 \)
5 \( 1 + (0.264 + 2.22i)T \)
7 \( 1 + (-0.698 + 2.55i)T \)
good11 \( 1 + (-0.371 + 0.643i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.05 + 2.05i)T + 13iT^{2} \)
17 \( 1 + (-1.69 - 6.33i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (0.946 + 1.63i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.11 + 1.36i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + 9.69iT - 29T^{2} \)
31 \( 1 + (-2.96 - 1.71i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.691 - 2.58i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + 0.817iT - 41T^{2} \)
43 \( 1 + (-1.59 + 1.59i)T - 43iT^{2} \)
47 \( 1 + (4.54 + 1.21i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (1.29 + 4.81i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (1.27 - 2.20i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.25 + 3.03i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-13.2 + 3.54i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 16.0T + 71T^{2} \)
73 \( 1 + (8.54 - 2.29i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-5.70 + 3.29i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-9.23 - 9.23i)T + 83iT^{2} \)
89 \( 1 + (-3.01 - 5.22i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.16 - 3.16i)T - 97iT^{2} \)
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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

−10.17271421960920114363312133949, −9.635846391939292785779882900294, −8.218062759194904358455210068757, −8.042707169352245594873411448806, −6.46585741938227189563449605258, −5.37708372570445585851181626634, −4.38174901680482678948506109323, −3.68916847839034924150356518109, −1.99735540798851866062670092075, −0.63912672160233422288413676949, 2.25499030152367403283208101337, 3.37791840329323706937634042055, 4.71041736087213991449566621008, 5.62781269314423466237424561692, 6.60373501480968128263134326396, 7.34735821890764515695356734560, 8.183156889051140844543100369602, 9.286808771379665600961629661695, 9.896837360369346966002954351807, 11.11050142492503018434466546376

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