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Properties

Label 2-63-7.2-c5-0-11
Degree $2$
Conductor $63$
Sign $-0.132 + 0.991i$
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  + (−2.54 + 4.40i)2-s + (3.08 + 5.33i)4-s + (20.9 − 36.2i)5-s + (−127. − 25.2i)7-s − 193.·8-s + (106. + 184. i)10-s + (36.0 + 62.4i)11-s − 632.·13-s + (434. − 495. i)14-s + (394. − 683. i)16-s + (−987. − 1.71e3i)17-s + (932. − 1.61e3i)19-s + 257.·20-s − 366.·22-s + (206. − 358. i)23-s + ⋯
L(s)  = 1  + (−0.449 + 0.778i)2-s + (0.0963 + 0.166i)4-s + (0.374 − 0.647i)5-s + (−0.980 − 0.194i)7-s − 1.07·8-s + (0.336 + 0.582i)10-s + (0.0898 + 0.155i)11-s − 1.03·13-s + (0.592 − 0.675i)14-s + (0.385 − 0.667i)16-s + (−0.829 − 1.43i)17-s + (0.592 − 1.02i)19-s + 0.144·20-s − 0.161·22-s + (0.0815 − 0.141i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.132 + 0.991i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.132 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.132 + 0.991i$
Analytic conductor: \(10.1041\)
Root analytic conductor: \(3.17870\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :5/2),\ -0.132 + 0.991i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.231780 - 0.264765i\)
\(L(\frac12)\) \(\approx\) \(0.231780 - 0.264765i\)
\(L(\frac{7}{2})\) 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 \( 1 + (127. + 25.2i)T \)
good2 \( 1 + (2.54 - 4.40i)T + (-16 - 27.7i)T^{2} \)
5 \( 1 + (-20.9 + 36.2i)T + (-1.56e3 - 2.70e3i)T^{2} \)
11 \( 1 + (-36.0 - 62.4i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 + 632.T + 3.71e5T^{2} \)
17 \( 1 + (987. + 1.71e3i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (-932. + 1.61e3i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (-206. + 358. i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + 731.T + 2.05e7T^{2} \)
31 \( 1 + (3.06e3 + 5.30e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (5.17e3 - 8.96e3i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 - 3.52e3T + 1.15e8T^{2} \)
43 \( 1 + 1.45e4T + 1.47e8T^{2} \)
47 \( 1 + (1.07e4 - 1.85e4i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (-6.28e3 - 1.08e4i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (1.80e4 + 3.12e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (2.01e3 - 3.48e3i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (7.78e3 + 1.34e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + 1.21e4T + 1.80e9T^{2} \)
73 \( 1 + (9.79e3 + 1.69e4i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (1.80e4 - 3.12e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 - 2.45e4T + 3.93e9T^{2} \)
89 \( 1 + (3.51e4 - 6.08e4i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 - 1.05e5T + 8.58e9T^{2} \)
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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

−13.60313141736047300224875638822, −12.66925079140809371636724775686, −11.53409189820901215731998301775, −9.612213936171578514783608211178, −9.098347115431822718935336626382, −7.47352532906445551643652992223, −6.59963245529876785773131606849, −5.00947575547540185369129840272, −2.85446907406613355330191360649, −0.17387037174013408892591958288, 1.98629366673045850786259681431, 3.34937414094138546247709352559, 5.80111432001345467078370952337, 6.86895219639836137764741346849, 8.818165706125496889832270657627, 10.00006840313375839032572918113, 10.54776121950410922423217440207, 11.92844222642365824583954889327, 12.86646995168249289393100286301, 14.36978427380195443985950356024

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