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Properties

Label 2-630-105.2-c1-0-6
Degree $2$
Conductor $630$
Sign $0.912 - 0.409i$
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  + (0.258 + 0.965i)2-s + (−0.866 + 0.499i)4-s + (−2.22 + 0.227i)5-s + (1.18 − 2.36i)7-s + (−0.707 − 0.707i)8-s + (−0.795 − 2.08i)10-s + (1.10 − 0.636i)11-s + (2.15 − 2.15i)13-s + (2.59 + 0.537i)14-s + (0.500 − 0.866i)16-s + (5.80 + 1.55i)17-s + (6.20 + 3.58i)19-s + (1.81 − 1.30i)20-s + (0.900 + 0.900i)22-s + (−4.14 + 1.11i)23-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s + (−0.433 + 0.249i)4-s + (−0.994 + 0.101i)5-s + (0.449 − 0.893i)7-s + (−0.249 − 0.249i)8-s + (−0.251 − 0.660i)10-s + (0.332 − 0.191i)11-s + (0.596 − 0.596i)13-s + (0.692 + 0.143i)14-s + (0.125 − 0.216i)16-s + (1.40 + 0.376i)17-s + (1.42 + 0.822i)19-s + (0.405 − 0.292i)20-s + (0.191 + 0.191i)22-s + (−0.864 + 0.231i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.912 - 0.409i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.912 - 0.409i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.912 - 0.409i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.912 - 0.409i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.42187 + 0.304809i\)
\(L(\frac12)\) \(\approx\) \(1.42187 + 0.304809i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 - 0.965i)T \)
3 \( 1 \)
5 \( 1 + (2.22 - 0.227i)T \)
7 \( 1 + (-1.18 + 2.36i)T \)
good11 \( 1 + (-1.10 + 0.636i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.15 + 2.15i)T - 13iT^{2} \)
17 \( 1 + (-5.80 - 1.55i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-6.20 - 3.58i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.14 - 1.11i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 1.25T + 29T^{2} \)
31 \( 1 + (2.35 + 4.08i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.99 - 0.535i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 0.655iT - 41T^{2} \)
43 \( 1 + (-7.20 + 7.20i)T - 43iT^{2} \)
47 \( 1 + (-2.80 - 10.4i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-3.55 + 13.2i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-0.688 - 1.19i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.57 - 4.46i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.12 + 4.19i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 0.159iT - 71T^{2} \)
73 \( 1 + (-9.90 - 2.65i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (9.23 + 5.33i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (6.09 + 6.09i)T + 83iT^{2} \)
89 \( 1 + (-3.79 + 6.57i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-10.9 - 10.9i)T + 97iT^{2} \)
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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

−10.61974182351100371896312296014, −9.835138256073864876151901587046, −8.569078585936024924695661957907, −7.66605501592443047338459426174, −7.52328447882179769904009342969, −6.12583869411009859825347608232, −5.23370703769035478869620665979, −3.92442920928176887944129410703, −3.47660047911764005673170106334, −1.00144237195915450426852081820, 1.22696977177783757529947107870, 2.80989044325434875284034648782, 3.81240382978707089818980717818, 4.86539174386239332140151774495, 5.73771076013078551985044525742, 7.12046217146979754075663376557, 8.041191595434816911359700999633, 8.912611447926152015640648737116, 9.613108249540152748062979423348, 10.77880238331969175750697249410

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