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Properties

Label 2-63-9.4-c1-0-3
Degree $2$
Conductor $63$
Sign $0.5 + 0.866i$
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  + (−0.673 − 1.16i)2-s + (1.70 − 0.300i)3-s + (0.0923 − 0.160i)4-s + (−1.26 + 2.19i)5-s + (−1.49 − 1.78i)6-s + (−0.5 − 0.866i)7-s − 2.94·8-s + (2.81 − 1.02i)9-s + 3.41·10-s + (−0.233 − 0.405i)11-s + (0.109 − 0.300i)12-s + (−2.91 + 5.04i)13-s + (−0.673 + 1.16i)14-s + (−1.5 + 4.12i)15-s + (1.79 + 3.11i)16-s + 3.87·17-s + ⋯
L(s)  = 1  + (−0.476 − 0.825i)2-s + (0.984 − 0.173i)3-s + (0.0461 − 0.0800i)4-s + (−0.566 + 0.980i)5-s + (−0.612 − 0.729i)6-s + (−0.188 − 0.327i)7-s − 1.04·8-s + (0.939 − 0.342i)9-s + 1.07·10-s + (−0.0705 − 0.122i)11-s + (0.0316 − 0.0868i)12-s + (−0.807 + 1.39i)13-s + (−0.180 + 0.311i)14-s + (−0.387 + 1.06i)15-s + (0.449 + 0.778i)16-s + 0.940·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.5 + 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.5 + 0.866i$
Analytic conductor: \(0.503057\)
Root analytic conductor: \(0.709265\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :1/2),\ 0.5 + 0.866i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.754133 - 0.435399i\)
\(L(\frac12)\) \(\approx\) \(0.754133 - 0.435399i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.70 + 0.300i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (0.673 + 1.16i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (1.26 - 2.19i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.233 + 0.405i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.91 - 5.04i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 3.87T + 17T^{2} \)
19 \( 1 + 2.18T + 19T^{2} \)
23 \( 1 + (-0.0530 + 0.0918i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.39 + 7.60i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.84 + 6.65i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 7.68T + 37T^{2} \)
41 \( 1 + (-1.11 + 1.92i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.613 + 1.06i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.66 - 4.61i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 0.716T + 53T^{2} \)
59 \( 1 + (0.368 - 0.637i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.479 + 0.829i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.81 + 8.34i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 13.2T + 71T^{2} \)
73 \( 1 + 10.2T + 73T^{2} \)
79 \( 1 + (-6.31 - 10.9i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.36 - 2.36i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 8.11T + 89T^{2} \)
97 \( 1 + (-6.80 - 11.7i)T + (-48.5 + 84.0i)T^{2} \)
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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

−14.74136465303592868309824437185, −13.88027788009327946045370078592, −12.31010736905681270687955969454, −11.31863124537251123039268213012, −10.13179156561009136394893239268, −9.300025567364884638193498655487, −7.75112478353975874046891473992, −6.60933675217119692794223740061, −3.77827692863151457379500076016, −2.34767761225658643415147265346, 3.22511432294268499379528251578, 5.19788020819904888395008645238, 7.22735312914029273535233301002, 8.188695489686515485120025030087, 8.876095289468611857697051455483, 10.16131892573071792915809126011, 12.28916140940002607108971259968, 12.76706954420576090456725363213, 14.51049208976836894755538779096, 15.37584204586849953575466606069

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