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Properties

Label 2-63-7.4-c5-0-2
Degree $2$
Conductor $63$
Sign $-0.431 + 0.902i$
Analytic cond. $10.1041$
Root an. cond. $3.17870$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (5.31 + 9.19i)2-s + (−40.4 + 70.0i)4-s + (−33.7 − 58.4i)5-s + (−120. − 48.3i)7-s − 518.·8-s + (358. − 620. i)10-s + (−261. + 452. i)11-s + 76.6·13-s + (−193. − 1.36e3i)14-s + (−1.46e3 − 2.53e3i)16-s + (−634. + 1.09e3i)17-s + (946. + 1.63e3i)19-s + 5.45e3·20-s − 5.55e3·22-s + (575. + 995. i)23-s + ⋯
L(s)  = 1  + (0.938 + 1.62i)2-s + (−1.26 + 2.18i)4-s + (−0.603 − 1.04i)5-s + (−0.927 − 0.373i)7-s − 2.86·8-s + (1.13 − 1.96i)10-s + (−0.651 + 1.12i)11-s + 0.125·13-s + (−0.264 − 1.85i)14-s + (−1.42 − 2.47i)16-s + (−0.532 + 0.922i)17-s + (0.601 + 1.04i)19-s + 3.04·20-s − 2.44·22-s + (0.226 + 0.392i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.431 + 0.902i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.431 + 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.431 + 0.902i$
Analytic conductor: \(10.1041\)
Root analytic conductor: \(3.17870\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :5/2),\ -0.431 + 0.902i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.458314 - 0.726909i\)
\(L(\frac12)\) \(\approx\) \(0.458314 - 0.726909i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (120. + 48.3i)T \)
good2 \( 1 + (-5.31 - 9.19i)T + (-16 + 27.7i)T^{2} \)
5 \( 1 + (33.7 + 58.4i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (261. - 452. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 - 76.6T + 3.71e5T^{2} \)
17 \( 1 + (634. - 1.09e3i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (-946. - 1.63e3i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (-575. - 995. i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + 3.85e3T + 2.05e7T^{2} \)
31 \( 1 + (-5.20e3 + 9.01e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (-2.80e3 - 4.85e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 + 1.42e4T + 1.15e8T^{2} \)
43 \( 1 + 1.48e4T + 1.47e8T^{2} \)
47 \( 1 + (-5.77e3 - 1.00e4i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (-2.33e3 + 4.05e3i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (1.45e4 - 2.52e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (5.92e3 + 1.02e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-1.81e4 + 3.13e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + 1.34e4T + 1.80e9T^{2} \)
73 \( 1 + (1.30e3 - 2.25e3i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (-3.91e4 - 6.78e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 - 1.67e4T + 3.93e9T^{2} \)
89 \( 1 + (1.83e4 + 3.18e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 - 3.61e4T + 8.58e9T^{2} \)
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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

−15.10794116990830141703533933176, −13.52795675979752935398348817989, −12.90784329519050183164086307811, −12.09211254470523033951469051661, −9.726903321524565315317125020540, −8.313631228617597119834786971848, −7.44601866164978588417762108449, −6.13349278545672327840794355681, −4.79841869661637857084929768211, −3.76952127395953585647105708599, 0.28655212983406735446375631174, 2.77364839949014680663238920139, 3.37438777597830535738007355923, 5.17712099431459114061882479255, 6.70261627687463777516566955716, 8.978878853389119796126368033379, 10.29083576003623120208454833576, 11.14433180065794673534123064835, 11.90265694395089440167795258187, 13.20449084102411292027066365045

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