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Properties

Label 2-63-63.58-c1-0-1
Degree $2$
Conductor $63$
Sign $0.367 - 0.930i$
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  − 0.495·2-s + (−0.221 + 1.71i)3-s − 1.75·4-s + (1.84 + 3.19i)5-s + (0.109 − 0.851i)6-s + (0.926 − 2.47i)7-s + 1.86·8-s + (−2.90 − 0.760i)9-s + (−0.915 − 1.58i)10-s + (0.446 − 0.772i)11-s + (0.388 − 3.01i)12-s + (0.598 − 1.03i)13-s + (−0.459 + 1.22i)14-s + (−5.90 + 2.46i)15-s + 2.58·16-s + (−0.124 − 0.216i)17-s + ⋯
L(s)  = 1  − 0.350·2-s + (−0.127 + 0.991i)3-s − 0.877·4-s + (0.825 + 1.43i)5-s + (0.0447 − 0.347i)6-s + (0.350 − 0.936i)7-s + 0.658·8-s + (−0.967 − 0.253i)9-s + (−0.289 − 0.501i)10-s + (0.134 − 0.233i)11-s + (0.112 − 0.869i)12-s + (0.165 − 0.287i)13-s + (−0.122 + 0.328i)14-s + (−1.52 + 0.636i)15-s + 0.646·16-s + (−0.0303 − 0.0525i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.367 - 0.930i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.367 - 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.367 - 0.930i$
Analytic conductor: \(0.503057\)
Root analytic conductor: \(0.709265\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :1/2),\ 0.367 - 0.930i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.584376 + 0.397501i\)
\(L(\frac12)\) \(\approx\) \(0.584376 + 0.397501i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.221 - 1.71i)T \)
7 \( 1 + (-0.926 + 2.47i)T \)
good2 \( 1 + 0.495T + 2T^{2} \)
5 \( 1 + (-1.84 - 3.19i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.446 + 0.772i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.598 + 1.03i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.124 + 0.216i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.40 + 2.43i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.23 + 2.14i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.07 - 3.58i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 3.58T + 31T^{2} \)
37 \( 1 + (2.36 - 4.09i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.39 - 4.14i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.98 + 8.64i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 10.1T + 47T^{2} \)
53 \( 1 + (4.94 + 8.56i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 1.81T + 59T^{2} \)
61 \( 1 - 10.8T + 61T^{2} \)
67 \( 1 - 1.02T + 67T^{2} \)
71 \( 1 + 4.94T + 71T^{2} \)
73 \( 1 + (0.915 + 1.58i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 1.79T + 79T^{2} \)
83 \( 1 + (-6.16 - 10.6i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.20 - 2.08i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.52 - 9.56i)T + (-48.5 + 84.0i)T^{2} \)
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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

−14.94015345075359193515938427746, −14.16531111457771363758729357207, −13.48066748328364788111040072536, −11.28651400855122974274482759828, −10.33699776654343802417257097408, −9.834857324660325024273639358441, −8.369070518855432246156323270726, −6.67629646032978749244009365691, −5.02725687679474812737482165602, −3.43742014098811917873730442880, 1.58907409519659394128148289584, 4.91231592321559048897814000150, 5.94026064560954979798465437727, 8.048225514206001504499001749918, 8.807043759177717069086755333363, 9.756734263498303097633538972330, 11.79226126339258664514117061881, 12.68473870850641259497094470096, 13.46798938686112438982235278152, 14.39142983010008203803237713066

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