Properties

Label 2-63-7.3-c2-0-4
Degree $2$
Conductor $63$
Sign $-0.386 + 0.922i$
Analytic cond. $1.71662$
Root an. cond. $1.31020$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 − 1.73i)2-s + (3 − 1.73i)5-s + (−3.5 − 6.06i)7-s − 8·8-s + (−6 − 3.46i)10-s + (5 − 8.66i)11-s + 12.1i·13-s + (−7 + 12.1i)14-s + (8 + 13.8i)16-s + (6 + 3.46i)17-s + (28.5 − 16.4i)19-s − 20·22-s + (20 + 34.6i)23-s + (−6.5 + 11.2i)25-s + (21 − 12.1i)26-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (0.600 − 0.346i)5-s + (−0.5 − 0.866i)7-s − 8-s + (−0.600 − 0.346i)10-s + (0.454 − 0.787i)11-s + 0.932i·13-s + (−0.5 + 0.866i)14-s + (0.5 + 0.866i)16-s + (0.352 + 0.203i)17-s + (1.5 − 0.866i)19-s − 0.909·22-s + (0.869 + 1.50i)23-s + (−0.260 + 0.450i)25-s + (0.807 − 0.466i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.386 + 0.922i$
Analytic conductor: \(1.71662\)
Root analytic conductor: \(1.31020\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :1),\ -0.386 + 0.922i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.550145 - 0.827060i\)
\(L(\frac12)\) \(\approx\) \(0.550145 - 0.827060i\)
\(L(2)\) 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 \( 1 + (3.5 + 6.06i)T \)
good2 \( 1 + (1 + 1.73i)T + (-2 + 3.46i)T^{2} \)
5 \( 1 + (-3 + 1.73i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-5 + 8.66i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 - 12.1iT - 169T^{2} \)
17 \( 1 + (-6 - 3.46i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-28.5 + 16.4i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-20 - 34.6i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + 16T + 841T^{2} \)
31 \( 1 + (-4.5 - 2.59i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (2.5 + 4.33i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 24.2iT - 1.68e3T^{2} \)
43 \( 1 + 19T + 1.84e3T^{2} \)
47 \( 1 + (-45 + 25.9i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (16 - 27.7i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (36 + 20.7i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-18 + 10.3i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (29.5 - 51.0i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 26T + 5.04e3T^{2} \)
73 \( 1 + (16.5 + 9.52i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (23.5 + 40.7i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 24.2iT - 6.88e3T^{2} \)
89 \( 1 + (102 - 58.8i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 48.4iT - 9.40e3T^{2} \)
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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

−14.04155575772000697237772865330, −13.29864896912335229859344788035, −11.79601124933233342885795450907, −10.97634116785294090208064560864, −9.655342920557803296324139533624, −9.134148080496635979086126326722, −7.10202213102005220828370999701, −5.62909091070315445450855641686, −3.40400231964604647511321347250, −1.25149660201208419863402630597, 2.91051489452654255124263516881, 5.57287677313616133854472190198, 6.62489361510316927389053354926, 7.87505601997023072789746656633, 9.187758856964400880568009922931, 10.08696605483776754907859360005, 11.90969665181931988904984600347, 12.74998536901454311563234903744, 14.36388323686347471253844365465, 15.20682151561462205079073779621

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