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Properties

Label 2-63-63.4-c1-0-1
Degree $2$
Conductor $63$
Sign $-0.398 - 0.917i$
Analytic cond. $0.503057$
Root an. cond. $0.709265$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.02 + 1.77i)2-s + (1.09 + 1.33i)3-s + (−1.10 − 1.92i)4-s − 0.146·5-s + (−3.50 + 0.582i)6-s + (0.0802 − 2.64i)7-s + 0.446·8-s + (−0.580 + 2.94i)9-s + (0.150 − 0.260i)10-s + 1.66·11-s + (1.34 − 3.59i)12-s + (0.0999 − 0.173i)13-s + (4.62 + 2.85i)14-s + (−0.160 − 0.195i)15-s + (1.75 − 3.04i)16-s + (3.13 − 5.43i)17-s + ⋯
L(s)  = 1  + (−0.726 + 1.25i)2-s + (0.635 + 0.772i)3-s + (−0.554 − 0.960i)4-s − 0.0654·5-s + (−1.43 + 0.237i)6-s + (0.0303 − 0.999i)7-s + 0.157·8-s + (−0.193 + 0.981i)9-s + (0.0474 − 0.0822i)10-s + 0.501·11-s + (0.389 − 1.03i)12-s + (0.0277 − 0.0480i)13-s + (1.23 + 0.763i)14-s + (−0.0415 − 0.0505i)15-s + (0.439 − 0.761i)16-s + (0.760 − 1.31i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.398 - 0.917i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.398 - 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.398 - 0.917i$
Analytic conductor: \(0.503057\)
Root analytic conductor: \(0.709265\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :1/2),\ -0.398 - 0.917i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.405946 + 0.618922i\)
\(L(\frac12)\) \(\approx\) \(0.405946 + 0.618922i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.09 - 1.33i)T \)
7 \( 1 + (-0.0802 + 2.64i)T \)
good2 \( 1 + (1.02 - 1.77i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + 0.146T + 5T^{2} \)
11 \( 1 - 1.66T + 11T^{2} \)
13 \( 1 + (-0.0999 + 0.173i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.13 + 5.43i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.45 - 5.99i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 6.18T + 23T^{2} \)
29 \( 1 + (2.46 + 4.27i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.25 - 2.18i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.50 + 6.06i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.15 + 2.00i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.940 + 1.62i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.905 + 1.56i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.67 - 4.62i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.28 - 3.95i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.339 + 0.587i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.09 - 5.35i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 1.27T + 71T^{2} \)
73 \( 1 + (0.778 - 1.34i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.39 - 11.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.75 - 6.50i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-4.53 - 7.85i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.98 + 6.90i)T + (-48.5 + 84.0i)T^{2} \)
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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

−15.66011471647795382754755924799, −14.25436494625811100775792821384, −13.99231497563107827972561435727, −11.84213999370821224359804797787, −10.12908250040292121045358014824, −9.486806805595607092866394750593, −8.067808701662464726541746406457, −7.35838418335006246666425713224, −5.62219236254586599863310090848, −3.79638942535556379219707184292, 1.84551670397262700885447495873, 3.35491205243251235752031185409, 6.13776616064728358828523901529, 7.990418862375061729641244595551, 8.956295253369637854279071252734, 9.877953628249155703499582721670, 11.50846374174293514911385971568, 12.14030403488318848521076017774, 13.13510777851168016813173627030, 14.51250605169663955248286782800

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