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Properties

Label 2-630-105.104-c1-0-9
Degree $2$
Conductor $630$
Sign $0.924 - 0.381i$
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  + 2-s + 4-s + 2.23·5-s + (2.23 + 1.41i)7-s + 8-s + 2.23·10-s + 5.65i·11-s − 4.47·13-s + (2.23 + 1.41i)14-s + 16-s − 3.16i·17-s − 3.16i·19-s + 2.23·20-s + 5.65i·22-s − 4·23-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + 0.999·5-s + (0.845 + 0.534i)7-s + 0.353·8-s + 0.707·10-s + 1.70i·11-s − 1.24·13-s + (0.597 + 0.377i)14-s + 0.250·16-s − 0.766i·17-s − 0.725i·19-s + 0.499·20-s + 1.20i·22-s − 0.834·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.924 - 0.381i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.924 - 0.381i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.924 - 0.381i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (629, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.924 - 0.381i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.69117 + 0.533455i\)
\(L(\frac12)\) \(\approx\) \(2.69117 + 0.533455i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
5 \( 1 - 2.23T \)
7 \( 1 + (-2.23 - 1.41i)T \)
good11 \( 1 - 5.65iT - 11T^{2} \)
13 \( 1 + 4.47T + 13T^{2} \)
17 \( 1 + 3.16iT - 17T^{2} \)
19 \( 1 + 3.16iT - 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 - 2.82iT - 29T^{2} \)
31 \( 1 + 6.32iT - 31T^{2} \)
37 \( 1 + 9.89iT - 37T^{2} \)
41 \( 1 - 4.47T + 41T^{2} \)
43 \( 1 + 1.41iT - 43T^{2} \)
47 \( 1 - 9.48iT - 47T^{2} \)
53 \( 1 + 4T + 53T^{2} \)
59 \( 1 - 4.47T + 59T^{2} \)
61 \( 1 + 9.48iT - 61T^{2} \)
67 \( 1 + 7.07iT - 67T^{2} \)
71 \( 1 - 1.41iT - 71T^{2} \)
73 \( 1 + 13.4T + 73T^{2} \)
79 \( 1 - 6T + 79T^{2} \)
83 \( 1 + 12.6iT - 83T^{2} \)
89 \( 1 + 4.47T + 89T^{2} \)
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

−10.70201058409333674478617821189, −9.691138381297599457495373048396, −9.233889968363438147026578209707, −7.69808345209293169729816753338, −7.10306119727941394578761724137, −5.92532432213345924448587459810, −4.99867483495500440700587231756, −4.48700744847924670532704838955, −2.53729219040747498920489622418, −1.97262706154105634666040171701, 1.45964813753227800410321395936, 2.75424634498949823268765854774, 3.99996257556866707717010121083, 5.14284742937600333533072463455, 5.84406582796620350396285208086, 6.74158417924385731892418082251, 7.929899630714025739821677234472, 8.641772789650263812203621401218, 10.01991682925018728654152842446, 10.49375375441422653495594020105

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