Properties

Label 2-630-315.202-c1-0-39
Degree $2$
Conductor $630$
Sign $-0.985 - 0.167i$
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 + (−1.28 + 1.16i)3-s + (−0.866 − 0.499i)4-s + (0.946 − 2.02i)5-s + (0.789 + 1.54i)6-s + (−1.09 − 2.40i)7-s + (−0.707 + 0.707i)8-s + (0.302 − 2.98i)9-s + (−1.71 − 1.43i)10-s + (1.34 + 2.32i)11-s + (1.69 − 0.363i)12-s + (−6.36 + 1.70i)13-s + (−2.60 + 0.436i)14-s + (1.13 + 3.70i)15-s + (0.500 + 0.866i)16-s + (−1.16 + 1.16i)17-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + (−0.741 + 0.670i)3-s + (−0.433 − 0.249i)4-s + (0.423 − 0.906i)5-s + (0.322 + 0.629i)6-s + (−0.414 − 0.909i)7-s + (−0.249 + 0.249i)8-s + (0.100 − 0.994i)9-s + (−0.541 − 0.454i)10-s + (0.405 + 0.701i)11-s + (0.488 − 0.104i)12-s + (−1.76 + 0.473i)13-s + (−0.697 + 0.116i)14-s + (0.293 + 0.955i)15-s + (0.125 + 0.216i)16-s + (−0.283 + 0.283i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 - 0.167i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.985 - 0.167i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.985 - 0.167i$
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: \((2,\ 630,\ (\ :1/2),\ -0.985 - 0.167i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0389878 + 0.463541i\)
\(L(\frac12)\) \(\approx\) \(0.0389878 + 0.463541i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.28 - 1.16i)T \)
5 \( 1 + (-0.946 + 2.02i)T \)
7 \( 1 + (1.09 + 2.40i)T \)
good11 \( 1 + (-1.34 - 2.32i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (6.36 - 1.70i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (1.16 - 1.16i)T - 17iT^{2} \)
19 \( 1 - 2.59T + 19T^{2} \)
23 \( 1 + (1.94 + 7.27i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (8.02 - 4.63i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.01 + 0.583i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.346 + 0.346i)T + 37iT^{2} \)
41 \( 1 + (8.19 + 4.73i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.101 + 0.0271i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (6.41 + 1.71i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (5.41 - 5.41i)T - 53iT^{2} \)
59 \( 1 + (0.290 - 0.503i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-11.4 + 6.62i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-8.62 + 2.31i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 5.20T + 71T^{2} \)
73 \( 1 + (6.71 + 6.71i)T + 73iT^{2} \)
79 \( 1 + (-2.22 + 1.28i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.89 + 10.8i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 - 1.13T + 89T^{2} \)
97 \( 1 + (-8.78 - 2.35i)T + (84.0 + 48.5i)T^{2} \)
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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

−9.966044847189718492915111516665, −9.721945429088016521696908775567, −8.846578724541960285444107964997, −7.32284992903061560377032180601, −6.39796820736258703226809158205, −5.09018644189292667960873103474, −4.62220482516440647830352781619, −3.66893467331115055501025068281, −1.91015027516717461547923297362, −0.24738648302552577863816162055, 2.17554487285203885227676619468, 3.35216014918047114354126644093, 5.20299900246044970808569243939, 5.66513550412129539670990419661, 6.57796475389735174654145487231, 7.28971366600324661290267667487, 8.079682536546051261628554450865, 9.511915077752687040217435304383, 9.942689598733146392375330590197, 11.44301854364490767925418550258

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