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Properties

Label 2-630-35.12-c1-0-4
Degree $2$
Conductor $630$
Sign $0.415 - 0.909i$
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  + (−0.965 + 0.258i)2-s + (0.866 − 0.499i)4-s + (1.38 + 1.75i)5-s + (2.58 + 0.559i)7-s + (−0.707 + 0.707i)8-s + (−1.79 − 1.33i)10-s + (1.83 + 3.17i)11-s + (−0.830 − 0.830i)13-s + (−2.64 + 0.128i)14-s + (0.500 − 0.866i)16-s + (0.761 + 0.204i)17-s + (−1.09 + 1.89i)19-s + (2.07 + 0.830i)20-s + (−2.59 − 2.59i)22-s + (−1.21 − 4.54i)23-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (0.433 − 0.249i)4-s + (0.618 + 0.785i)5-s + (0.977 + 0.211i)7-s + (−0.249 + 0.249i)8-s + (−0.566 − 0.423i)10-s + (0.553 + 0.958i)11-s + (−0.230 − 0.230i)13-s + (−0.706 + 0.0343i)14-s + (0.125 − 0.216i)16-s + (0.184 + 0.0494i)17-s + (−0.251 + 0.434i)19-s + (0.464 + 0.185i)20-s + (−0.553 − 0.553i)22-s + (−0.253 − 0.947i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.415 - 0.909i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.415 - 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.415 - 0.909i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (397, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.415 - 0.909i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11895 + 0.718815i\)
\(L(\frac12)\) \(\approx\) \(1.11895 + 0.718815i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 - 0.258i)T \)
3 \( 1 \)
5 \( 1 + (-1.38 - 1.75i)T \)
7 \( 1 + (-2.58 - 0.559i)T \)
good11 \( 1 + (-1.83 - 3.17i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.830 + 0.830i)T + 13iT^{2} \)
17 \( 1 + (-0.761 - 0.204i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (1.09 - 1.89i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.21 + 4.54i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + 2.62iT - 29T^{2} \)
31 \( 1 + (-0.0359 + 0.0207i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.248 - 0.0664i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 8.98iT - 41T^{2} \)
43 \( 1 + (0.474 - 0.474i)T - 43iT^{2} \)
47 \( 1 + (1.65 + 6.18i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-7.64 - 2.04i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-5.35 - 9.27i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.72 - 0.996i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.71 - 6.39i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 8.11T + 71T^{2} \)
73 \( 1 + (-2.55 + 9.52i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-11.6 - 6.70i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (9.73 + 9.73i)T + 83iT^{2} \)
89 \( 1 + (-0.715 + 1.23i)T + (-44.5 - 77.0i)T^{2} \)
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

−10.45612897138695716532854253621, −10.02106485879510028842970428651, −9.042681039788405842859403938540, −8.114518748398371002886658356577, −7.27116531589033467485942275279, −6.43613326959905081781959683431, −5.47495489696348953566040548193, −4.27590059466530035268071266758, −2.60421717837181026570780493793, −1.62668413371298413934532213010, 1.00997864481205690381397040070, 2.11812145018474310328442749794, 3.74503714177167090812220135948, 4.98175804794313969375416628431, 5.87996569575860188447119562288, 7.03370907012087561112172117896, 8.075506810752672509835089316340, 8.768943740054739811227734676489, 9.420983861385756235340484917872, 10.39793800840927833232814698556

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