Properties

Degree $2$
Conductor $63$
Sign $-0.0832 - 0.996i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.23 + 2.13i)2-s + (−1.73 − 0.0789i)3-s + (−2.02 + 3.51i)4-s + (1.29 − 2.24i)5-s + (−1.96 − 3.78i)6-s + (0.5 + 0.866i)7-s − 5.05·8-s + (2.98 + 0.273i)9-s + 6.38·10-s + (−2.25 − 3.90i)11-s + (3.78 − 5.91i)12-s + (−0.5 + 0.866i)13-s + (−1.23 + 2.13i)14-s + (−2.42 + 3.78i)15-s + (−2.16 − 3.74i)16-s − 0.945·17-s + ⋯
L(s)  = 1  + (0.869 + 1.50i)2-s + (−0.998 − 0.0455i)3-s + (−1.01 + 1.75i)4-s + (0.579 − 1.00i)5-s + (−0.800 − 1.54i)6-s + (0.188 + 0.327i)7-s − 1.78·8-s + (0.995 + 0.0910i)9-s + 2.01·10-s + (−0.680 − 1.17i)11-s + (1.09 − 1.70i)12-s + (−0.138 + 0.240i)13-s + (−0.328 + 0.569i)14-s + (−0.625 + 0.977i)15-s + (−0.540 − 0.936i)16-s − 0.229·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0832 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0832 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.0832 - 0.996i$
Motivic weight: \(1\)
Character: $\chi_{63} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :1/2),\ -0.0832 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.702852 + 0.764047i\)
\(L(\frac12)\) \(\approx\) \(0.702852 + 0.764047i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.73 + 0.0789i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + (-1.23 - 2.13i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-1.29 + 2.24i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.25 + 3.90i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 0.945T + 17T^{2} \)
19 \( 1 + 4.05T + 19T^{2} \)
23 \( 1 + (-0.136 + 0.236i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.23 + 2.13i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.16 - 2.01i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 1.78T + 37T^{2} \)
41 \( 1 + (-3.20 + 5.54i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.21 - 9.03i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-6.08 - 10.5i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 6.27T + 53T^{2} \)
59 \( 1 + (-1.36 + 2.36i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.13 - 1.96i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.90 + 13.6i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.27T + 71T^{2} \)
73 \( 1 + 1.50T + 73T^{2} \)
79 \( 1 + (7.35 + 12.7i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.472 - 0.819i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 14.3T + 89T^{2} \)
97 \( 1 + (-5.74 - 9.95i)T + (-48.5 + 84.0i)T^{2} \)
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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

−15.57530185650260208974041344855, −14.17270306976103270469413050116, −13.10109112154208255982556037977, −12.54434365159366506156971212450, −10.98420286801384286058802832411, −9.082978284771635242721427468663, −7.86115365477267947192291609274, −6.27440148941157616623710342386, −5.52690236278043528756984142182, −4.51715697813269216610725851510, 2.25819598315759132741083676837, 4.25055572786579143279363424997, 5.53236216195131829681392088431, 7.02949503817024933281254373927, 9.868050420658947220552197659345, 10.48656646886926510309189015504, 11.22405540566764182357493756630, 12.45536476668167191634047494839, 13.17608208582805332798130305479, 14.41025686264992557738414229539

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