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Properties

Label 2-630-63.5-c1-0-9
Degree $2$
Conductor $630$
Sign $0.0281 - 0.999i$
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  + i·2-s + (−1.61 + 0.630i)3-s − 4-s + (0.5 + 0.866i)5-s + (−0.630 − 1.61i)6-s + (2.57 − 0.594i)7-s − i·8-s + (2.20 − 2.03i)9-s + (−0.866 + 0.5i)10-s + (0.676 + 0.390i)11-s + (1.61 − 0.630i)12-s + (2.26 + 1.31i)13-s + (0.594 + 2.57i)14-s + (−1.35 − 1.08i)15-s + 16-s + (−3.02 − 5.24i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.931 + 0.363i)3-s − 0.5·4-s + (0.223 + 0.387i)5-s + (−0.257 − 0.658i)6-s + (0.974 − 0.224i)7-s − 0.353i·8-s + (0.735 − 0.677i)9-s + (−0.273 + 0.158i)10-s + (0.204 + 0.117i)11-s + (0.465 − 0.181i)12-s + (0.629 + 0.363i)13-s + (0.158 + 0.689i)14-s + (−0.349 − 0.279i)15-s + 0.250·16-s + (−0.734 − 1.27i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0281 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0281 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.0281 - 0.999i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.0281 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.901413 + 0.876397i\)
\(L(\frac12)\) \(\approx\) \(0.901413 + 0.876397i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (1.61 - 0.630i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-2.57 + 0.594i)T \)
good11 \( 1 + (-0.676 - 0.390i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.26 - 1.31i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.02 + 5.24i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-6.09 - 3.51i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.99 + 2.30i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.96 + 2.28i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 7.48iT - 31T^{2} \)
37 \( 1 + (5.42 - 9.40i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.46 - 9.47i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.45 + 7.70i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 1.00T + 47T^{2} \)
53 \( 1 + (8.30 - 4.79i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 2.64T + 59T^{2} \)
61 \( 1 - 2.13iT - 61T^{2} \)
67 \( 1 - 12.0T + 67T^{2} \)
71 \( 1 + 10.1iT - 71T^{2} \)
73 \( 1 + (-7.67 + 4.42i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 15.9T + 79T^{2} \)
83 \( 1 + (-2.03 - 3.52i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-3.78 + 6.56i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.00 - 2.88i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.80003184985252921940034365566, −9.978048696074775981979637848818, −9.100694855092320095638430062337, −8.076654026065973598708448782318, −6.94483942537057528148336745605, −6.49828106048382406147718749375, −5.14571182056949460691151790487, −4.79938799960000620395614234040, −3.41467953238512353605480722877, −1.27040063474372753555392297030, 0.999438473682143257886601855930, 2.05485739574119600549248458006, 3.80026162869117208134533382264, 4.99248998807220240753116446180, 5.53390062882327195193536546511, 6.70737364296151320496063011641, 7.87039061783621656071881273647, 8.697499468084081104976929498873, 9.642939176491687158371830436281, 10.82199775586094580260213780109

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