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Properties

Label 2-630-315.202-c1-0-29
Degree $2$
Conductor $630$
Sign $0.880 + 0.473i$
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 + (1.17 + 1.27i)3-s + (−0.866 − 0.499i)4-s + (2.11 − 0.712i)5-s + (1.53 − 0.800i)6-s + (−2.06 − 1.65i)7-s + (−0.707 + 0.707i)8-s + (−0.259 + 2.98i)9-s + (−0.139 − 2.23i)10-s + (1.96 + 3.41i)11-s + (−0.375 − 1.69i)12-s + (5.47 − 1.46i)13-s + (−2.12 + 1.56i)14-s + (3.39 + 1.87i)15-s + (0.500 + 0.866i)16-s + (4.02 − 4.02i)17-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + (0.675 + 0.737i)3-s + (−0.433 − 0.249i)4-s + (0.947 − 0.318i)5-s + (0.627 − 0.326i)6-s + (−0.781 − 0.624i)7-s + (−0.249 + 0.249i)8-s + (−0.0864 + 0.996i)9-s + (−0.0440 − 0.705i)10-s + (0.593 + 1.02i)11-s + (−0.108 − 0.488i)12-s + (1.51 − 0.406i)13-s + (−0.569 + 0.419i)14-s + (0.875 + 0.483i)15-s + (0.125 + 0.216i)16-s + (0.975 − 0.975i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.880 + 0.473i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.880 + 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.880 + 0.473i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (517, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.880 + 0.473i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.17345 - 0.547675i\)
\(L(\frac12)\) \(\approx\) \(2.17345 - 0.547675i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.17 - 1.27i)T \)
5 \( 1 + (-2.11 + 0.712i)T \)
7 \( 1 + (2.06 + 1.65i)T \)
good11 \( 1 + (-1.96 - 3.41i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-5.47 + 1.46i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (-4.02 + 4.02i)T - 17iT^{2} \)
19 \( 1 + 4.50T + 19T^{2} \)
23 \( 1 + (-0.162 - 0.605i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (-1.29 + 0.746i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.958 - 0.553i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.36 - 3.36i)T + 37iT^{2} \)
41 \( 1 + (9.58 + 5.53i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.640 - 0.171i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (-0.489 - 0.131i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (7.48 - 7.48i)T - 53iT^{2} \)
59 \( 1 + (-6.31 + 10.9i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.632 + 0.364i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (11.6 - 3.13i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 5.34T + 71T^{2} \)
73 \( 1 + (-8.09 - 8.09i)T + 73iT^{2} \)
79 \( 1 + (8.12 - 4.69i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.743 + 2.77i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + 10.1T + 89T^{2} \)
97 \( 1 + (12.1 + 3.26i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.24333421895418011293570780235, −9.838252721927941978959718194941, −9.111125980445003158629604952984, −8.265168238058504758718438755688, −6.87961043724806993651202283188, −5.80861325080874017916470506516, −4.69480613148776795811054910637, −3.79411130511031539207250257119, −2.82142635928746863618903438556, −1.44123226402586678655609571705, 1.50130779244122904952772519423, 3.00756215751458529785240748796, 3.81733882450112023354611140198, 5.80706099069475137386543815398, 6.20869952937994389116578022641, 6.76634965217148932784085456999, 8.255196569781231944791181244872, 8.706465393341901496732621262994, 9.456577619186918238458858912716, 10.50941033106019595930365727683

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