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Properties

Label 2-63-9.5-c2-0-3
Degree $2$
Conductor $63$
Sign $-0.297 - 0.954i$
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  + (−2.95 + 1.70i)2-s + (2.25 + 1.98i)3-s + (3.80 − 6.58i)4-s + (6.84 + 3.95i)5-s + (−10.0 − 2.01i)6-s + (−1.32 − 2.29i)7-s + 12.2i·8-s + (1.14 + 8.92i)9-s − 26.9·10-s + (−13.1 + 7.58i)11-s + (21.6 − 7.29i)12-s + (2.78 − 4.81i)13-s + (7.80 + 4.50i)14-s + (7.58 + 22.4i)15-s + (−5.71 − 9.90i)16-s − 18.1i·17-s + ⋯
L(s)  = 1  + (−1.47 + 0.851i)2-s + (0.750 + 0.660i)3-s + (0.950 − 1.64i)4-s + (1.36 + 0.790i)5-s + (−1.67 − 0.335i)6-s + (−0.188 − 0.327i)7-s + 1.53i·8-s + (0.126 + 0.991i)9-s − 2.69·10-s + (−1.19 + 0.689i)11-s + (1.80 − 0.607i)12-s + (0.214 − 0.370i)13-s + (0.557 + 0.321i)14-s + (0.505 + 1.49i)15-s + (−0.357 − 0.619i)16-s − 1.06i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.297 - 0.954i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.297 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.297 - 0.954i$
Analytic conductor: \(1.71662\)
Root analytic conductor: \(1.31020\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (50, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :1),\ -0.297 - 0.954i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.515948 + 0.700839i\)
\(L(\frac12)\) \(\approx\) \(0.515948 + 0.700839i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.25 - 1.98i)T \)
7 \( 1 + (1.32 + 2.29i)T \)
good2 \( 1 + (2.95 - 1.70i)T + (2 - 3.46i)T^{2} \)
5 \( 1 + (-6.84 - 3.95i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (13.1 - 7.58i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-2.78 + 4.81i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + 18.1iT - 289T^{2} \)
19 \( 1 - 14.4T + 361T^{2} \)
23 \( 1 + (12.4 + 7.19i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-12.4 + 7.18i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-7.54 + 13.0i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 - 15.6T + 1.36e3T^{2} \)
41 \( 1 + (-26.5 - 15.3i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (23.5 + 40.8i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-54.9 + 31.7i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 55.0iT - 2.80e3T^{2} \)
59 \( 1 + (58.5 + 33.8i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (7.27 + 12.6i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (1.84 - 3.18i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 88.8iT - 5.04e3T^{2} \)
73 \( 1 - 41.4T + 5.32e3T^{2} \)
79 \( 1 + (-8.98 - 15.5i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (102. - 59.0i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 11.3iT - 7.92e3T^{2} \)
97 \( 1 + (-25.9 - 44.9i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−15.36357905143829645195371372846, −14.26189429698381595681082374987, −13.38032074925705662875038458197, −10.72905141344240204606087379686, −9.990856033585045233659984275430, −9.490900724046276569195472367238, −8.021792096108657439882716892062, −6.98562570488258901543339840039, −5.47040258409455291396220349071, −2.48017828772347767696512467280, 1.41865404538516189710975712539, 2.74686564295684364934616634904, 5.95502299836292480823257880962, 7.84544272661330081388473915591, 8.780169526431444913874585995412, 9.532113674208890164790897958154, 10.58297140463909447989711994304, 12.18240104755456160214774726822, 13.07547516179790437166502773590, 13.98457277413086611358518060935

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