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Properties

Label 2-63-9.4-c3-0-15
Degree $2$
Conductor $63$
Sign $-0.306 + 0.952i$
Analytic cond. $3.71712$
Root an. cond. $1.92798$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.231 − 0.400i)2-s + (1.24 − 5.04i)3-s + (3.89 − 6.74i)4-s + (−2.26 + 3.92i)5-s + (−2.30 + 0.665i)6-s + (−3.5 − 6.06i)7-s − 7.29·8-s + (−23.8 − 12.6i)9-s + 2.09·10-s + (3.34 + 5.78i)11-s + (−29.1 − 28.0i)12-s + (17.5 − 30.3i)13-s + (−1.61 + 2.80i)14-s + (16.9 + 16.3i)15-s + (−29.4 − 51.0i)16-s + 94.5·17-s + ⋯
L(s)  = 1  + (−0.0817 − 0.141i)2-s + (0.240 − 0.970i)3-s + (0.486 − 0.842i)4-s + (−0.202 + 0.350i)5-s + (−0.157 + 0.0453i)6-s + (−0.188 − 0.327i)7-s − 0.322·8-s + (−0.884 − 0.466i)9-s + 0.0662·10-s + (0.0915 + 0.158i)11-s + (−0.701 − 0.674i)12-s + (0.374 − 0.647i)13-s + (−0.0308 + 0.0535i)14-s + (0.291 + 0.280i)15-s + (−0.460 − 0.797i)16-s + 1.34·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.306 + 0.952i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.306 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.306 + 0.952i$
Analytic conductor: \(3.71712\)
Root analytic conductor: \(1.92798\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ -0.306 + 0.952i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.867829 - 1.19050i\)
\(L(\frac12)\) \(\approx\) \(0.867829 - 1.19050i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.24 + 5.04i)T \)
7 \( 1 + (3.5 + 6.06i)T \)
good2 \( 1 + (0.231 + 0.400i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 + (2.26 - 3.92i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-3.34 - 5.78i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-17.5 + 30.3i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 94.5T + 4.91e3T^{2} \)
19 \( 1 + 0.0567T + 6.85e3T^{2} \)
23 \( 1 + (22.0 - 38.2i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-122. - 211. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-113. + 196. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 264.T + 5.06e4T^{2} \)
41 \( 1 + (-8.06 + 13.9i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (33.4 + 57.9i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-59.0 - 102. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 588.T + 1.48e5T^{2} \)
59 \( 1 + (-419. + 726. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-97.8 - 169. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (334. - 579. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 889.T + 3.57e5T^{2} \)
73 \( 1 + 160.T + 3.89e5T^{2} \)
79 \( 1 + (-248. - 429. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-221. - 384. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 756.T + 7.04e5T^{2} \)
97 \( 1 + (-557. - 965. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−14.24480879498071935411891857423, −13.02194674496327415016676869233, −11.85877757926391107357963533952, −10.80607783998325184762178360544, −9.602161800681930657896106660329, −7.942034617902585668947410232867, −6.83768121999160406410829021425, −5.66974459099252904286877506706, −3.03508732191702789188761122589, −1.12934365307258046161278215150, 2.96246029510722675054489363908, 4.37496758329835527170524279580, 6.18244287623878995162014186681, 7.964954474723023447130655236103, 8.834066544094859714630454803944, 10.13736179191328767652284907512, 11.52532387311342206328077550303, 12.33639507988303355604305698753, 13.83855669838552621285645418200, 15.01090267609462247481482219709

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