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Properties

Label 2-63-7.4-c5-0-15
Degree $2$
Conductor $63$
Sign $-0.992 - 0.118i$
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  + (−0.395 − 0.684i)2-s + (15.6 − 27.1i)4-s + (−52.0 − 90.2i)5-s + (−7.12 + 129. i)7-s − 50.1·8-s + (−41.1 + 71.3i)10-s + (−248. + 430. i)11-s − 206.·13-s + (91.4 − 46.3i)14-s + (−482. − 835. i)16-s + (31.5 − 54.6i)17-s + (−661. − 1.14e3i)19-s − 3.26e3·20-s + 393.·22-s + (−97.2 − 168. i)23-s + ⋯
L(s)  = 1  + (−0.0698 − 0.121i)2-s + (0.490 − 0.849i)4-s + (−0.931 − 1.61i)5-s + (−0.0549 + 0.998i)7-s − 0.276·8-s + (−0.130 + 0.225i)10-s + (−0.620 + 1.07i)11-s − 0.338·13-s + (0.124 − 0.0631i)14-s + (−0.470 − 0.815i)16-s + (0.0265 − 0.0459i)17-s + (−0.420 − 0.728i)19-s − 1.82·20-s + 0.173·22-s + (−0.0383 − 0.0663i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.118i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.992 - 0.118i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.992 - 0.118i$
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: \((2,\ 63,\ (\ :5/2),\ -0.992 - 0.118i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.0376668 + 0.635526i\)
\(L(\frac12)\) \(\approx\) \(0.0376668 + 0.635526i\)
\(L(\frac{7}{2})\) 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 \( 1 + (7.12 - 129. i)T \)
good2 \( 1 + (0.395 + 0.684i)T + (-16 + 27.7i)T^{2} \)
5 \( 1 + (52.0 + 90.2i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (248. - 430. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + 206.T + 3.71e5T^{2} \)
17 \( 1 + (-31.5 + 54.6i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (661. + 1.14e3i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (97.2 + 168. i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + 4.32e3T + 2.05e7T^{2} \)
31 \( 1 + (-3.76e3 + 6.51e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (5.17e3 + 8.96e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 - 4.18e3T + 1.15e8T^{2} \)
43 \( 1 - 5.96e3T + 1.47e8T^{2} \)
47 \( 1 + (2.19e3 + 3.80e3i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (-8.89e3 + 1.54e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (-1.75e3 + 3.03e3i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (-5.31e3 - 9.20e3i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-6.63e3 + 1.14e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + 3.88e4T + 1.80e9T^{2} \)
73 \( 1 + (1.56e4 - 2.71e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (1.97e4 + 3.42e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 - 1.02e5T + 3.93e9T^{2} \)
89 \( 1 + (5.64e4 + 9.77e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 - 3.03e4T + 8.58e9T^{2} \)
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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

−13.03362918956244499770032342795, −12.23544701157219287371818916515, −11.34824807947215310715652561805, −9.732479535830584873842220635518, −8.804170476816982752584267187656, −7.45543869112104456208578026632, −5.58807260012321450387952585079, −4.60778866363151414139165423396, −2.09211284816022871516737471710, −0.28400935131352758421345348680, 2.92349361251216726866432611491, 3.83149634142669973452798908691, 6.45217917745580103600456006529, 7.41229740351126897557598976279, 8.180321088817603791552714983958, 10.43706585978135634408303443707, 11.04413882974535540259722188144, 12.09424435501793983936283550861, 13.55202788767945386986015260962, 14.61797927900795289993661054232

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