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Properties

Label 2-63-21.17-c5-0-3
Degree $2$
Conductor $63$
Sign $0.595 - 0.803i$
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  + (2.44 − 1.41i)2-s + (−12.0 + 20.7i)4-s + (−25.7 − 44.5i)5-s + (122.5 + 42.4i)7-s + 158. i·8-s + (−126 − 72.7i)10-s + (500. + 289. i)11-s + 1.15e3i·13-s + (360. − 69.2i)14-s + (−159. − 277. i)16-s + (−188. + 326. i)17-s + (−1.50e3 + 866. i)19-s + 1.23e3·20-s + 1.63e3·22-s + (3.99e3 − 2.30e3i)23-s + ⋯
L(s)  = 1  + (0.433 − 0.249i)2-s + (−0.375 + 0.649i)4-s + (−0.460 − 0.796i)5-s + (0.944 + 0.327i)7-s + 0.874i·8-s + (−0.398 − 0.230i)10-s + (1.24 + 0.720i)11-s + 1.89i·13-s + (0.490 − 0.0944i)14-s + (−0.156 − 0.270i)16-s + (−0.158 + 0.274i)17-s + (−0.954 + 0.550i)19-s + 0.690·20-s + 0.720·22-s + (1.57 − 0.908i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.595 - 0.803i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.595 - 0.803i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.595 - 0.803i$
Analytic conductor: \(10.1041\)
Root analytic conductor: \(3.17870\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :5/2),\ 0.595 - 0.803i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.70318 + 0.856984i\)
\(L(\frac12)\) \(\approx\) \(1.70318 + 0.856984i\)
\(L(\frac{7}{2})\) 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 \( 1 + (-122.5 - 42.4i)T \)
good2 \( 1 + (-2.44 + 1.41i)T + (16 - 27.7i)T^{2} \)
5 \( 1 + (25.7 + 44.5i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (-500. - 289. i)T + (8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 - 1.15e3iT - 3.71e5T^{2} \)
17 \( 1 + (188. - 326. i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (1.50e3 - 866. i)T + (1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (-3.99e3 + 2.30e3i)T + (3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 - 4.60e3iT - 2.05e7T^{2} \)
31 \( 1 + (1.96e3 + 1.13e3i)T + (1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (101.5 + 175. i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 + 85.7T + 1.15e8T^{2} \)
43 \( 1 - 4.69e3T + 1.47e8T^{2} \)
47 \( 1 + (1.19e4 + 2.07e4i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (6.42e3 + 3.70e3i)T + (2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (2.48e3 - 4.30e3i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (1.50e4 - 8.71e3i)T + (4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (1.50e4 - 2.61e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 - 3.96e3iT - 1.80e9T^{2} \)
73 \( 1 + (-4.61e4 - 2.66e4i)T + (1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (-1.96e4 - 3.41e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 - 6.35e4T + 3.93e9T^{2} \)
89 \( 1 + (5.26e4 + 9.12e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 + 2.06e3iT - 8.58e9T^{2} \)
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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

−14.21758828311570533232130888397, −12.73824659573494337871810893282, −12.05316010042322467250814973901, −11.22148101185462249431897287498, −9.035359296455537233954367865253, −8.547205162984101544439511941446, −6.91784798929355510636316461200, −4.75049121435581100521974186356, −4.13227855018354116295642424793, −1.79522513458509439042459382811, 0.871947835499704179834162005396, 3.45411691602373278205225265708, 4.94272085306200569905742017015, 6.31832827073590899144843151306, 7.64686319162157859929268942938, 9.105713196988070020110662839779, 10.67065147965992413405712718766, 11.26966820683500959781979590375, 12.97308076772562938397559847324, 14.01119799571522626789624173865

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