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Properties

Label 2-63-63.20-c1-0-4
Degree $2$
Conductor $63$
Sign $-0.692 + 0.721i$
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  + (−1.97 − 1.13i)2-s + (0.578 − 1.63i)3-s + (1.59 + 2.75i)4-s + (−0.717 − 1.24i)5-s + (−2.99 + 2.55i)6-s + (−2.40 − 1.11i)7-s − 2.69i·8-s + (−2.33 − 1.88i)9-s + 3.26i·10-s + (2.80 + 1.61i)11-s + (5.41 − 1.00i)12-s + (4.43 − 2.55i)13-s + (3.47 + 4.92i)14-s + (−2.44 + 0.451i)15-s + (0.119 − 0.207i)16-s + 1.09·17-s + ⋯
L(s)  = 1  + (−1.39 − 0.804i)2-s + (0.334 − 0.942i)3-s + (0.795 + 1.37i)4-s + (−0.320 − 0.555i)5-s + (−1.22 + 1.04i)6-s + (−0.907 − 0.419i)7-s − 0.951i·8-s + (−0.776 − 0.629i)9-s + 1.03i·10-s + (0.844 + 0.487i)11-s + (1.56 − 0.289i)12-s + (1.22 − 0.709i)13-s + (0.927 + 1.31i)14-s + (−0.630 + 0.116i)15-s + (0.0298 − 0.0517i)16-s + 0.264·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.692 + 0.721i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.692 + 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.692 + 0.721i$
Analytic conductor: \(0.503057\)
Root analytic conductor: \(0.709265\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :1/2),\ -0.692 + 0.721i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.186541 - 0.437924i\)
\(L(\frac12)\) \(\approx\) \(0.186541 - 0.437924i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.578 + 1.63i)T \)
7 \( 1 + (2.40 + 1.11i)T \)
good2 \( 1 + (1.97 + 1.13i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (0.717 + 1.24i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.80 - 1.61i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-4.43 + 2.55i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 1.09T + 17T^{2} \)
19 \( 1 - 4.48iT - 19T^{2} \)
23 \( 1 + (-3.47 + 2.00i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.02 - 0.593i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.24 - 1.87i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 0.239T + 37T^{2} \)
41 \( 1 + (-3.71 - 6.43i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.82 - 6.62i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.11 - 3.65i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 7.01iT - 53T^{2} \)
59 \( 1 + (4.73 + 8.20i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.82 + 1.63i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.330 + 0.571i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 3.82iT - 71T^{2} \)
73 \( 1 - 7.31iT - 73T^{2} \)
79 \( 1 + (1.83 - 3.16i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.45 - 9.44i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 13.6T + 89T^{2} \)
97 \( 1 + (-2.69 - 1.55i)T + (48.5 + 84.0i)T^{2} \)
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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

−14.43774371613803619824094158064, −12.94297198642855350628004629658, −12.31681668555976572621189403781, −11.09589945699888816021294856616, −9.788409594005608981836469380657, −8.740360628311118463668635822796, −7.85615946311581143094563245081, −6.46436863352392689320904694421, −3.34091255397121325083077074527, −1.16972861696127013472767891170, 3.53786021103306902518950838853, 6.00566203157387147764646895058, 7.16224305504598902343677629083, 8.894644153988514565837566317656, 9.117169602911260789621143059951, 10.50808529411448498470970372816, 11.43418172654709211579379068554, 13.57302218270449218147251419517, 14.91724853030596000616581729866, 15.64183342901212976672163941506

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