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Properties

Label 2-630-5.4-c1-0-3
Degree $2$
Conductor $630$
Sign $0.994 + 0.100i$
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 − 4-s + (−2.22 − 0.224i)5-s + i·7-s + i·8-s + (−0.224 + 2.22i)10-s + 4.89·11-s + 4.44i·13-s + 14-s + 16-s + 2i·17-s − 1.55·19-s + (2.22 + 0.224i)20-s − 4.89i·22-s − 2.89i·23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + (−0.994 − 0.100i)5-s + 0.377i·7-s + 0.353i·8-s + (−0.0710 + 0.703i)10-s + 1.47·11-s + 1.23i·13-s + 0.267·14-s + 0.250·16-s + 0.485i·17-s − 0.355·19-s + (0.497 + 0.0502i)20-s − 1.04i·22-s − 0.604i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.100i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.994 + 0.100i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.994 + 0.100i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20180 - 0.0605493i\)
\(L(\frac12)\) \(\approx\) \(1.20180 - 0.0605493i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + (2.22 + 0.224i)T \)
7 \( 1 - iT \)
good11 \( 1 - 4.89T + 11T^{2} \)
13 \( 1 - 4.44iT - 13T^{2} \)
17 \( 1 - 2iT - 17T^{2} \)
19 \( 1 + 1.55T + 19T^{2} \)
23 \( 1 + 2.89iT - 23T^{2} \)
29 \( 1 - 6.89T + 29T^{2} \)
31 \( 1 - 8.89T + 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 - 1.10T + 41T^{2} \)
43 \( 1 + 0.898iT - 43T^{2} \)
47 \( 1 - 8.89iT - 47T^{2} \)
53 \( 1 - 10.8iT - 53T^{2} \)
59 \( 1 + 1.55T + 59T^{2} \)
61 \( 1 - 3.55T + 61T^{2} \)
67 \( 1 - 8iT - 67T^{2} \)
71 \( 1 - 1.10T + 71T^{2} \)
73 \( 1 - 2.89iT - 73T^{2} \)
79 \( 1 + 6.89T + 79T^{2} \)
83 \( 1 - 2.44iT - 83T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 + 15.7iT - 97T^{2} \)
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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

−10.77978238130661553106619261712, −9.688977376503746855911810381446, −8.830133923242571812937321381777, −8.305228659001904633933983976304, −6.96789027740596121622536605099, −6.17813385027328162864953002477, −4.47739934615752819725415685874, −4.13835302570792377367435308883, −2.78941120750453085865894528983, −1.26269572151845981901787625124, 0.822428553646264225555685138781, 3.17361220245073802465020795395, 4.11105245719994294565629021426, 5.04091120199844640166316057104, 6.37534852595493371583771746251, 7.00375233718295929533320522989, 8.008271270560157847940488043327, 8.557246996831482733398238683821, 9.679831163742516411236608743851, 10.52249599462199763648347994259

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