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Properties

Label 2-630-35.4-c1-0-15
Degree $2$
Conductor $630$
Sign $0.324 + 0.946i$
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  + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (2.23 − 0.133i)5-s + (−0.866 − 2.5i)7-s − 0.999i·8-s + (1.86 − 1.23i)10-s − 2i·13-s + (−2 − 1.73i)14-s + (−0.5 − 0.866i)16-s + (1.73 + i)17-s + (−1 − 1.73i)19-s + (0.999 − 1.99i)20-s + (0.866 − 0.5i)23-s + (4.96 − 0.598i)25-s + (−1 − 1.73i)26-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (0.998 − 0.0599i)5-s + (−0.327 − 0.944i)7-s − 0.353i·8-s + (0.590 − 0.389i)10-s − 0.554i·13-s + (−0.534 − 0.462i)14-s + (−0.125 − 0.216i)16-s + (0.420 + 0.242i)17-s + (−0.229 − 0.397i)19-s + (0.223 − 0.447i)20-s + (0.180 − 0.104i)23-s + (0.992 − 0.119i)25-s + (−0.196 − 0.339i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.324 + 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.324 + 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.324 + 0.946i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.324 + 0.946i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.91649 - 1.36928i\)
\(L(\frac12)\) \(\approx\) \(1.91649 - 1.36928i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 \)
5 \( 1 + (-2.23 + 0.133i)T \)
7 \( 1 + (0.866 + 2.5i)T \)
good11 \( 1 + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 + (-1.73 - i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1 + 1.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.866 + 0.5i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + T + 29T^{2} \)
31 \( 1 + (5 - 8.66i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-6.92 + 4i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 3T + 41T^{2} \)
43 \( 1 - 5iT - 43T^{2} \)
47 \( 1 + (-6.92 + 4i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.19 + 3i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (1 - 1.73i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.5 - 7.79i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.06 + 3.5i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + (-8.66 - 5i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (5 + 8.66i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 9iT - 83T^{2} \)
89 \( 1 + (-3.5 - 6.06i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−10.53216214462286160476356729339, −9.783745975686101809926788947500, −8.917905033063397482896407686805, −7.58530134593419210693672132332, −6.67283762913732135692701177291, −5.79290821752424911697524805141, −4.88042480040498311830414139625, −3.72754148573961795538465383326, −2.62652323393542724311638625983, −1.17532756090815000519104581651, 1.97676035651480685141837753913, 2.99167260709427651732464272674, 4.36344574377281478588780753210, 5.57851149995383191972348638540, 6.00267055428814973770936797386, 6.98352900191776595433218816888, 8.085274543952564062568800201166, 9.198836590801732566440159019566, 9.658075549004584543433779415784, 10.86496704420777038541166655435

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