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Properties

Label 2-630-63.5-c1-0-1
Degree $2$
Conductor $630$
Sign $-0.924 - 0.380i$
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 + (−1.68 + 0.389i)3-s − 4-s + (−0.5 − 0.866i)5-s + (−0.389 − 1.68i)6-s + (0.408 + 2.61i)7-s − i·8-s + (2.69 − 1.31i)9-s + (0.866 − 0.5i)10-s + (3.42 + 1.97i)11-s + (1.68 − 0.389i)12-s + (2.82 + 1.63i)13-s + (−2.61 + 0.408i)14-s + (1.18 + 1.26i)15-s + 16-s + (−0.497 − 0.861i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.974 + 0.225i)3-s − 0.5·4-s + (−0.223 − 0.387i)5-s + (−0.159 − 0.688i)6-s + (0.154 + 0.987i)7-s − 0.353i·8-s + (0.898 − 0.438i)9-s + (0.273 − 0.158i)10-s + (1.03 + 0.596i)11-s + (0.487 − 0.112i)12-s + (0.784 + 0.452i)13-s + (−0.698 + 0.109i)14-s + (0.305 + 0.327i)15-s + 0.250·16-s + (−0.120 − 0.208i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.924 - 0.380i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.924 - 0.380i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.924 - 0.380i$
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: \((2,\ 630,\ (\ :1/2),\ -0.924 - 0.380i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.147992 + 0.748532i\)
\(L(\frac12)\) \(\approx\) \(0.147992 + 0.748532i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.68 - 0.389i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-0.408 - 2.61i)T \)
good11 \( 1 + (-3.42 - 1.97i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.82 - 1.63i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.497 + 0.861i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.90 + 2.83i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (6.67 - 3.85i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.10 - 2.36i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 5.58iT - 31T^{2} \)
37 \( 1 + (5.05 - 8.75i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5.16 + 8.94i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.10 - 8.83i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 4.73T + 47T^{2} \)
53 \( 1 + (-8.82 + 5.09i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 3.40T + 59T^{2} \)
61 \( 1 - 6.29iT - 61T^{2} \)
67 \( 1 - 9.22T + 67T^{2} \)
71 \( 1 - 2.74iT - 71T^{2} \)
73 \( 1 + (12.7 - 7.36i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 1.84T + 79T^{2} \)
83 \( 1 + (0.789 + 1.36i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-6.92 + 11.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.776 - 0.448i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.14600858276780576484663588361, −9.954850517729150013926005739823, −9.101423565403534412711644866654, −8.513727003552565764090748614620, −7.18244630029938067990515544041, −6.39582349549626485291479526069, −5.64720745341400301139784334745, −4.66483260124978666470414833065, −3.86427064947030307556218123844, −1.62368962594583689797749920109, 0.49863221508794197571842987044, 1.88900013787102302567329893640, 3.87334496055464655218036154839, 4.15958398834326106050226125692, 5.82673874610083875299103170727, 6.41045333833706097277384286128, 7.57730691971137650180038621965, 8.427054847708170594786750228535, 9.695871763060318674121605721123, 10.68096603025498857680244803617

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