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Properties

Label 2-63-9.4-c1-0-5
Degree $2$
Conductor $63$
Sign $-1$
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  + (−1.26 − 2.19i)2-s + (−1.11 − 1.32i)3-s + (−2.20 + 3.82i)4-s + (0.439 − 0.761i)5-s + (−1.49 + 4.12i)6-s + (−0.5 − 0.866i)7-s + 6.10·8-s + (−0.520 + 2.95i)9-s − 2.22·10-s + (−1.93 − 3.35i)11-s + (7.52 − 1.32i)12-s + (2.72 − 4.72i)13-s + (−1.26 + 2.19i)14-s + (−1.5 + 0.264i)15-s + (−3.31 − 5.74i)16-s + 1.65·17-s + ⋯
L(s)  = 1  + (−0.895 − 1.55i)2-s + (−0.642 − 0.766i)3-s + (−1.10 + 1.91i)4-s + (0.196 − 0.340i)5-s + (−0.612 + 1.68i)6-s + (−0.188 − 0.327i)7-s + 2.15·8-s + (−0.173 + 0.984i)9-s − 0.704·10-s + (−0.584 − 1.01i)11-s + (2.17 − 0.383i)12-s + (0.756 − 1.30i)13-s + (−0.338 + 0.586i)14-s + (−0.387 + 0.0682i)15-s + (−0.829 − 1.43i)16-s + 0.400·17-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-1$
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: \((2,\ 63,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.446501i\)
\(L(\frac12)\) \(\approx\) \(0.446501i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.11 + 1.32i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (1.26 + 2.19i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-0.439 + 0.761i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.93 + 3.35i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.72 + 4.72i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 1.65T + 17T^{2} \)
19 \( 1 - 2.41T + 19T^{2} \)
23 \( 1 + (1.58 - 2.73i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.02 - 5.23i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.27 + 3.94i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 4.55T + 37T^{2} \)
41 \( 1 + (-0.592 + 1.02i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.0923 + 0.160i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.511 - 0.885i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 7.29T + 53T^{2} \)
59 \( 1 + (3.33 - 5.76i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.29 - 2.24i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.47 + 2.56i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.68T + 71T^{2} \)
73 \( 1 + 12.7T + 73T^{2} \)
79 \( 1 + (-2.97 - 5.15i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.109 - 0.189i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 11.0T + 89T^{2} \)
97 \( 1 + (6.25 + 10.8i)T + (-48.5 + 84.0i)T^{2} \)
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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

−13.58087970425569155400583735761, −13.04395292847368247683733706430, −12.00028143698731483946229408241, −10.94827473610392361225938941475, −10.25689347726947070255953287165, −8.686994610037223371250459282559, −7.68956404391940840217284972890, −5.57769392948336853746921687801, −3.16036832803650895504663024812, −1.01109110029024279561629139939, 4.69576061943248052062180948784, 6.04850827985136318986794470493, 6.95699953408859447720521181886, 8.559220139825862720447198160512, 9.660558586882332550376923865121, 10.42492082544693444960938704206, 12.02265871013177196540307175108, 13.95712866616802025166411675990, 14.89957752099113417837853173708, 15.86791763878571522475499390565

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