Properties

Label 2-630-105.2-c1-0-5
Degree $2$
Conductor $630$
Sign $0.143 - 0.989i$
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 + (2.16 + 0.561i)5-s + (2.06 + 1.65i)7-s + (−0.707 − 0.707i)8-s + (0.0180 + 2.23i)10-s + (0.565 − 0.326i)11-s + (0.771 − 0.771i)13-s + (−1.06 + 2.42i)14-s + (0.500 − 0.866i)16-s + (−2.06 − 0.554i)17-s + (4.34 + 2.50i)19-s + (−2.15 + 0.596i)20-s + (0.461 + 0.461i)22-s + (0.905 − 0.242i)23-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s + (−0.433 + 0.249i)4-s + (0.967 + 0.251i)5-s + (0.779 + 0.626i)7-s + (−0.249 − 0.249i)8-s + (0.00570 + 0.707i)10-s + (0.170 − 0.0984i)11-s + (0.214 − 0.214i)13-s + (−0.285 + 0.647i)14-s + (0.125 − 0.216i)16-s + (−0.501 − 0.134i)17-s + (0.996 + 0.575i)19-s + (−0.481 + 0.133i)20-s + (0.0984 + 0.0984i)22-s + (0.188 − 0.0505i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.143 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.143 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.143 - 0.989i$
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: \((2,\ 630,\ (\ :1/2),\ 0.143 - 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.49466 + 1.29365i\)
\(L(\frac12)\) \(\approx\) \(1.49466 + 1.29365i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.16 - 0.561i)T \)
7 \( 1 + (-2.06 - 1.65i)T \)
good11 \( 1 + (-0.565 + 0.326i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.771 + 0.771i)T - 13iT^{2} \)
17 \( 1 + (2.06 + 0.554i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-4.34 - 2.50i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.905 + 0.242i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 9.79T + 29T^{2} \)
31 \( 1 + (0.897 + 1.55i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-6.26 + 1.67i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 7.89iT - 41T^{2} \)
43 \( 1 + (0.0657 - 0.0657i)T - 43iT^{2} \)
47 \( 1 + (0.854 + 3.18i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (1.91 - 7.13i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (3.39 + 5.87i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.33 - 7.51i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.66 + 6.21i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 7.68iT - 71T^{2} \)
73 \( 1 + (9.62 + 2.58i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-10.3 - 5.96i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.838 + 0.838i)T + 83iT^{2} \)
89 \( 1 + (-8.65 + 14.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (8.59 + 8.59i)T + 97iT^{2} \)
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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

−10.84838812126509619718713198403, −9.618714111709893543233653481553, −9.120771368202165261766492650357, −8.067176732858990449595090051900, −7.25261984642158468457637763427, −6.05489352569633320424801922397, −5.58015690991045548038981195954, −4.54639826330502954862172280500, −3.08551413009730621947070743926, −1.70856937670880723893518262668, 1.21161393274559243892031955572, 2.28047970178620591640436526799, 3.74342192049626539959849278730, 4.81450641721845201925314201840, 5.58579903478576242939603594223, 6.77762232160236759774385056597, 7.82743017429855926203279221247, 9.016497863870744627159369289116, 9.515381761716386825880058977076, 10.55456153259392593686286451084

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