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Properties

Label 2-630-315.202-c1-0-4
Degree $2$
Conductor $630$
Sign $0.185 - 0.982i$
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 + (−1.65 − 0.523i)3-s + (−0.866 − 0.499i)4-s + (2.21 + 0.313i)5-s + (−0.932 + 1.45i)6-s + (−2.14 + 1.54i)7-s + (−0.707 + 0.707i)8-s + (2.45 + 1.72i)9-s + (0.875 − 2.05i)10-s + (−0.481 − 0.834i)11-s + (1.16 + 1.27i)12-s + (−4.03 + 1.08i)13-s + (0.939 + 2.47i)14-s + (−3.49 − 1.67i)15-s + (0.500 + 0.866i)16-s + (−3.20 + 3.20i)17-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + (−0.953 − 0.302i)3-s + (−0.433 − 0.249i)4-s + (0.990 + 0.140i)5-s + (−0.380 + 0.595i)6-s + (−0.811 + 0.584i)7-s + (−0.249 + 0.249i)8-s + (0.817 + 0.575i)9-s + (0.276 − 0.650i)10-s + (−0.145 − 0.251i)11-s + (0.337 + 0.369i)12-s + (−1.11 + 0.299i)13-s + (0.251 + 0.661i)14-s + (−0.901 − 0.432i)15-s + (0.125 + 0.216i)16-s + (−0.777 + 0.777i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.185 - 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.185 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.185 - 0.982i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (517, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.185 - 0.982i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.386989 + 0.320683i\)
\(L(\frac12)\) \(\approx\) \(0.386989 + 0.320683i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.65 + 0.523i)T \)
5 \( 1 + (-2.21 - 0.313i)T \)
7 \( 1 + (2.14 - 1.54i)T \)
good11 \( 1 + (0.481 + 0.834i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (4.03 - 1.08i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (3.20 - 3.20i)T - 17iT^{2} \)
19 \( 1 + 8.00T + 19T^{2} \)
23 \( 1 + (-0.884 - 3.30i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (2.04 - 1.18i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-7.54 - 4.35i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.06 - 5.06i)T + 37iT^{2} \)
41 \( 1 + (-2.71 - 1.56i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (9.94 + 2.66i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (-0.195 - 0.0523i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (3.48 - 3.48i)T - 53iT^{2} \)
59 \( 1 + (-1.68 + 2.91i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.30 + 1.33i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.63 - 0.437i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 14.1T + 71T^{2} \)
73 \( 1 + (-0.754 - 0.754i)T + 73iT^{2} \)
79 \( 1 + (-11.2 + 6.52i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.94 - 14.7i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + 14.2T + 89T^{2} \)
97 \( 1 + (-6.29 - 1.68i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.71407362829947610672431521616, −10.13219637449004126609080484420, −9.383675919720666238842823353177, −8.364760997630577304936700421249, −6.73638355738317988599243915572, −6.29922831311627440254270218831, −5.33484145415952097641140537170, −4.40187241328432439004393890069, −2.74267810240517619402171226665, −1.78234255567908185550572338802, 0.27304006738954580848036361389, 2.51803339867844002756903307414, 4.29380361201733152687138707047, 4.88368811557414419227104909195, 6.06795608764111096012868549225, 6.55773725915000523116624753258, 7.38660616289909145214509079623, 8.796328846037750469274515860554, 9.776702713357976741787730558711, 10.14149274752464949809659454897

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