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Properties

Label 2-630-35.12-c1-0-5
Degree $2$
Conductor $630$
Sign $0.999 - 0.0275i$
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 + (1.79 − 1.33i)5-s + (−2.55 + 0.698i)7-s + (−0.707 + 0.707i)8-s + (−1.38 + 1.75i)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 + (6.33 + 1.69i)17-s + (0.946 − 1.63i)19-s + (0.880 − 2.05i)20-s + (−0.525 − 0.525i)22-s + (1.36 + 5.11i)23-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (0.433 − 0.249i)4-s + (0.800 − 0.599i)5-s + (−0.964 + 0.264i)7-s + (−0.249 + 0.249i)8-s + (−0.437 + 0.555i)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 + (1.53 + 0.411i)17-s + (0.217 − 0.375i)19-s + (0.196 − 0.459i)20-s + (−0.112 − 0.112i)22-s + (0.285 + 1.06i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0275i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0275i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.999 - 0.0275i$
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: \((2,\ 630,\ (\ :1/2),\ 0.999 - 0.0275i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20202 + 0.0165536i\)
\(L(\frac12)\) \(\approx\) \(1.20202 + 0.0165536i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.79 + 1.33i)T \)
7 \( 1 + (2.55 - 0.698i)T \)
good11 \( 1 + (-0.371 - 0.643i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.05 - 2.05i)T + 13iT^{2} \)
17 \( 1 + (-6.33 - 1.69i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-0.946 + 1.63i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.36 - 5.11i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + 9.69iT - 29T^{2} \)
31 \( 1 + (-2.96 + 1.71i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.58 + 0.691i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 0.817iT - 41T^{2} \)
43 \( 1 + (-1.59 + 1.59i)T - 43iT^{2} \)
47 \( 1 + (1.21 + 4.54i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-4.81 - 1.29i)T + (45.8 + 26.5i)T^{2} \)
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 \( 1 + (3.54 - 13.2i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 16.0T + 71T^{2} \)
73 \( 1 + (2.29 - 8.54i)T + (-63.2 - 36.5i)T^{2} \)
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

−10.08195061449687436715816278192, −9.783620998769961266455986392686, −9.006439796177181860561602724867, −8.135368675047478162346761650290, −7.04487237634878621062032831432, −6.05194381579477798968724884656, −5.50188085971849126539233776854, −3.93135661818560516592429897857, −2.52391863492926282631861118635, −1.11659055317809220023476540030, 1.12892687118478778412766127055, 2.82848696857390361777041108460, 3.48392153013817723898183641168, 5.35398682760239243581442479894, 6.28916396301660838114698129147, 6.99802260216160088597076835976, 8.014625893402428289212393012421, 9.065184945181073328517100193766, 9.842448065121511874425823810063, 10.41195052904949307149824672450

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