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Properties

Label 2-63-63.38-c1-0-3
Degree $2$
Conductor $63$
Sign $0.766 + 0.642i$
Analytic cond. $0.503057$
Root an. cond. $0.709265$
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  − 0.293i·2-s + (−1.65 − 0.518i)3-s + 1.91·4-s + (1.53 − 2.65i)5-s + (−0.152 + 0.485i)6-s + (−1.41 + 2.23i)7-s − 1.15i·8-s + (2.46 + 1.71i)9-s + (−0.778 − 0.449i)10-s + (−3.37 + 1.94i)11-s + (−3.16 − 0.993i)12-s + (−2.02 + 1.17i)13-s + (0.656 + 0.416i)14-s + (−3.90 + 3.58i)15-s + 3.48·16-s + (−1.68 + 2.91i)17-s + ⋯
L(s)  = 1  − 0.207i·2-s + (−0.954 − 0.299i)3-s + 0.956·4-s + (0.684 − 1.18i)5-s + (−0.0622 + 0.198i)6-s + (−0.535 + 0.844i)7-s − 0.406i·8-s + (0.820 + 0.571i)9-s + (−0.246 − 0.142i)10-s + (−1.01 + 0.587i)11-s + (−0.912 − 0.286i)12-s + (−0.562 + 0.324i)13-s + (0.175 + 0.111i)14-s + (−1.00 + 0.925i)15-s + 0.872·16-s + (−0.408 + 0.706i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.766 + 0.642i$
Analytic conductor: \(0.503057\)
Root analytic conductor: \(0.709265\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :1/2),\ 0.766 + 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.789108 - 0.287236i\)
\(L(\frac12)\) \(\approx\) \(0.789108 - 0.287236i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.65 + 0.518i)T \)
7 \( 1 + (1.41 - 2.23i)T \)
good2 \( 1 + 0.293iT - 2T^{2} \)
5 \( 1 + (-1.53 + 2.65i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (3.37 - 1.94i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.02 - 1.17i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.68 - 2.91i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.20 + 1.27i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.58 - 1.49i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.67 + 2.12i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 0.472iT - 31T^{2} \)
37 \( 1 + (3.89 + 6.74i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.12 - 5.41i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.06 + 3.57i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 4.05T + 47T^{2} \)
53 \( 1 + (-4.99 - 2.88i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 4.68T + 59T^{2} \)
61 \( 1 - 1.60iT - 61T^{2} \)
67 \( 1 - 1.57T + 67T^{2} \)
71 \( 1 + 13.6iT - 71T^{2} \)
73 \( 1 + (0.856 + 0.494i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + 9.27T + 79T^{2} \)
83 \( 1 + (-5.49 + 9.51i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (2.15 + 3.73i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.98 - 2.87i)T + (48.5 + 84.0i)T^{2} \)
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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

−15.26442731791682036315905369411, −13.10881749369898878251096184065, −12.63609700898989609253825538776, −11.70496719192488540064326276169, −10.40153860540498525943060510965, −9.288846789599110747513879231835, −7.46757349838368909353570910634, −6.06179552799599569360739800104, −5.09747791802124992974656455974, −2.06241003115688454590496600898, 3.00039875845733903013171643653, 5.45680076696329972759554524964, 6.62475830661277071773764655942, 7.37571900340829696696909684183, 9.953752903307945170536960757705, 10.60275541283919301518139920194, 11.37031778377928747430200946895, 12.86189340238821977191177340365, 14.12857254033226387288263045699, 15.33548301858673616280553698363

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