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Properties

Label 2-630-105.2-c1-0-12
Degree $2$
Conductor $630$
Sign $-0.929 + 0.369i$
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 + (−1.24 + 1.85i)5-s + (−1.31 − 2.29i)7-s + (0.707 + 0.707i)8-s + (2.11 + 0.719i)10-s + (1.05 − 0.608i)11-s + (2.42 − 2.42i)13-s + (−1.87 + 1.86i)14-s + (0.500 − 0.866i)16-s + (3.34 + 0.896i)17-s + (−7.32 − 4.22i)19-s + (0.146 − 2.23i)20-s + (−0.859 − 0.859i)22-s + (−7.15 + 1.91i)23-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s + (−0.433 + 0.249i)4-s + (−0.555 + 0.831i)5-s + (−0.497 − 0.867i)7-s + (0.249 + 0.249i)8-s + (0.669 + 0.227i)10-s + (0.317 − 0.183i)11-s + (0.672 − 0.672i)13-s + (−0.501 + 0.498i)14-s + (0.125 − 0.216i)16-s + (0.811 + 0.217i)17-s + (−1.68 − 0.970i)19-s + (0.0328 − 0.498i)20-s + (−0.183 − 0.183i)22-s + (−1.49 + 0.399i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 + 0.369i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.929 + 0.369i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.929 + 0.369i$
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: \((2,\ 630,\ (\ :1/2),\ -0.929 + 0.369i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.109359 - 0.570408i\)
\(L(\frac12)\) \(\approx\) \(0.109359 - 0.570408i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.24 - 1.85i)T \)
7 \( 1 + (1.31 + 2.29i)T \)
good11 \( 1 + (-1.05 + 0.608i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.42 + 2.42i)T - 13iT^{2} \)
17 \( 1 + (-3.34 - 0.896i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (7.32 + 4.22i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (7.15 - 1.91i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 7.86T + 29T^{2} \)
31 \( 1 + (3.47 + 6.01i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.19 + 1.12i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 5.75iT - 41T^{2} \)
43 \( 1 + (-1.25 + 1.25i)T - 43iT^{2} \)
47 \( 1 + (1.94 + 7.27i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-2.15 + 8.03i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-0.0764 - 0.132i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.62 + 11.4i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.50 - 13.0i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 11.6iT - 71T^{2} \)
73 \( 1 + (-7.69 - 2.06i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (1.09 + 0.634i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.08 - 3.08i)T + 83iT^{2} \)
89 \( 1 + (1.92 - 3.33i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.22 - 3.22i)T + 97iT^{2} \)
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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

−10.35454272315181927031746323439, −9.634280982049744510257975657406, −8.429330826439781942981839444409, −7.67352291070832508184662615614, −6.73525082022628443332798169766, −5.73308107891741593177996398340, −3.89732985129956636249253529526, −3.74480190735027786958144008025, −2.25048042600221328516039242205, −0.33414042549985868808402855824, 1.73541685859996439045098821638, 3.66672842514064280089553726736, 4.52527305514525879510991597469, 5.79451603964349735879213740794, 6.31843497403380291969858452737, 7.62716206462656273810507253525, 8.388628785913873709374529560175, 9.061890332108885712400741698836, 9.784457101476883368776086784345, 10.95095891875012916246367407971

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