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Properties

Label 2-630-105.2-c1-0-7
Degree $2$
Conductor $630$
Sign $0.840 + 0.542i$
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 + (1.48 + 1.67i)5-s + (1.30 − 2.30i)7-s + (0.707 + 0.707i)8-s + (1.23 − 1.86i)10-s + (3.21 − 1.85i)11-s + (−4.62 + 4.62i)13-s + (−2.56 − 0.662i)14-s + (0.500 − 0.866i)16-s + (5.06 + 1.35i)17-s + (−0.866 − 0.5i)19-s + (−2.12 − 0.707i)20-s + (−2.62 − 2.62i)22-s + (4.10 − 1.09i)23-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s + (−0.433 + 0.249i)4-s + (0.663 + 0.748i)5-s + (0.492 − 0.870i)7-s + (0.249 + 0.249i)8-s + (0.389 − 0.590i)10-s + (0.968 − 0.559i)11-s + (−1.28 + 1.28i)13-s + (−0.684 − 0.177i)14-s + (0.125 − 0.216i)16-s + (1.22 + 0.329i)17-s + (−0.198 − 0.114i)19-s + (−0.474 − 0.158i)20-s + (−0.559 − 0.559i)22-s + (0.854 − 0.229i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.840 + 0.542i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.840 + 0.542i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.840 + 0.542i$
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: \((2,\ 630,\ (\ :1/2),\ 0.840 + 0.542i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.51847 - 0.447621i\)
\(L(\frac12)\) \(\approx\) \(1.51847 - 0.447621i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.48 - 1.67i)T \)
7 \( 1 + (-1.30 + 2.30i)T \)
good11 \( 1 + (-3.21 + 1.85i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (4.62 - 4.62i)T - 13iT^{2} \)
17 \( 1 + (-5.06 - 1.35i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (0.866 + 0.5i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.10 + 1.09i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 2.82T + 29T^{2} \)
31 \( 1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-10.4 + 2.79i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 0.880iT - 41T^{2} \)
43 \( 1 + (-3 + 3i)T - 43iT^{2} \)
47 \( 1 + (-2.32 - 8.69i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (0.581 - 2.16i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (5.83 + 10.0i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.62 + 2.81i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.56 - 9.56i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 10.2iT - 71T^{2} \)
73 \( 1 + (13.9 + 3.74i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-2.38 - 1.37i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6.53 - 6.53i)T + 83iT^{2} \)
89 \( 1 + (4.94 - 8.57i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (9.24 + 9.24i)T + 97iT^{2} \)
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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

−10.59938090921911019783539575131, −9.656723269724921851003245979681, −9.188081186110901066846652014196, −7.81786565972850134982002646247, −7.05163649252767264676659925687, −6.07816850433543122173115654384, −4.72880292653376587705904861964, −3.77904875606644749474743056300, −2.54392219225770456374796043151, −1.28861114684287750746198181325, 1.23907272182159115839751284693, 2.76372674786876487565830839044, 4.58221377063638498597194088901, 5.29848184062854407323399278779, 5.98762595824786683554605677225, 7.24048888169448116789166990883, 8.058213035117390180818985566135, 8.949031978069033470377644629528, 9.633713431805765402068448270464, 10.29128915455658227900741816171

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