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Properties

Label 2-630-63.58-c1-0-25
Degree $2$
Conductor $630$
Sign $0.980 - 0.196i$
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  + 2-s + (1.01 + 1.39i)3-s + 4-s + (−0.5 − 0.866i)5-s + (1.01 + 1.39i)6-s + (1.85 − 1.88i)7-s + 8-s + (−0.919 + 2.85i)9-s + (−0.5 − 0.866i)10-s + (3.22 − 5.58i)11-s + (1.01 + 1.39i)12-s + (0.332 − 0.575i)13-s + (1.85 − 1.88i)14-s + (0.702 − 1.58i)15-s + 16-s + (0.411 + 0.713i)17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.588 + 0.808i)3-s + 0.5·4-s + (−0.223 − 0.387i)5-s + (0.416 + 0.571i)6-s + (0.699 − 0.714i)7-s + 0.353·8-s + (−0.306 + 0.951i)9-s + (−0.158 − 0.273i)10-s + (0.971 − 1.68i)11-s + (0.294 + 0.404i)12-s + (0.0920 − 0.159i)13-s + (0.494 − 0.505i)14-s + (0.181 − 0.408i)15-s + 0.250·16-s + (0.0998 + 0.173i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.196i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 - 0.196i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.980 - 0.196i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.980 - 0.196i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.84869 + 0.283168i\)
\(L(\frac12)\) \(\approx\) \(2.84869 + 0.283168i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (-1.01 - 1.39i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-1.85 + 1.88i)T \)
good11 \( 1 + (-3.22 + 5.58i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.332 + 0.575i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.411 - 0.713i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.77 - 4.80i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.74 - 3.02i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.03 - 3.52i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 7.63T + 31T^{2} \)
37 \( 1 + (5.32 - 9.21i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.511 + 0.886i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.15 + 7.18i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 2.60T + 47T^{2} \)
53 \( 1 + (3.27 + 5.67i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 2.91T + 59T^{2} \)
61 \( 1 + 0.767T + 61T^{2} \)
67 \( 1 - 7.28T + 67T^{2} \)
71 \( 1 - 3.44T + 71T^{2} \)
73 \( 1 + (-4.71 - 8.16i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 5.40T + 79T^{2} \)
83 \( 1 + (-0.897 - 1.55i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.06 - 3.57i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (8.98 + 15.5i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.85656349180487001308912266805, −9.862982007805103284024722728897, −8.591524108818511687311833865564, −8.304755065268727830291682125091, −7.07364273867020309845840590494, −5.79963829821907566499036718510, −4.96071179898361734772659325923, −3.80175686086231093448080811768, −3.45411475682614134425389559674, −1.55123536004718502711617717447, 1.78205269808436850985976313137, 2.57696876267286017804773835777, 3.97327215205800055039872795535, 4.90647283048265893065477990644, 6.26105307616188532650209077138, 6.99581383197639543604030141371, 7.66573346341292075541019308493, 8.813971962361508007590691842957, 9.480537791996401357470197007997, 10.88218707494605902496766745417

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