Properties

Label 2-63-63.58-c1-0-4
Degree $2$
Conductor $63$
Sign $0.997 - 0.0735i$
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  + 2.05·2-s + (−1.70 + 0.283i)3-s + 2.21·4-s + (0.0731 + 0.126i)5-s + (−3.50 + 0.582i)6-s + (−2.33 − 1.25i)7-s + 0.446·8-s + (2.83 − 0.969i)9-s + (0.150 + 0.260i)10-s + (−0.832 + 1.44i)11-s + (−3.78 + 0.628i)12-s + (0.0999 − 0.173i)13-s + (−4.78 − 2.57i)14-s + (−0.160 − 0.195i)15-s − 3.51·16-s + (3.13 + 5.43i)17-s + ⋯
L(s)  = 1  + 1.45·2-s + (−0.986 + 0.163i)3-s + 1.10·4-s + (0.0327 + 0.0566i)5-s + (−1.43 + 0.237i)6-s + (−0.880 − 0.473i)7-s + 0.157·8-s + (0.946 − 0.323i)9-s + (0.0474 + 0.0822i)10-s + (−0.250 + 0.434i)11-s + (−1.09 + 0.181i)12-s + (0.0277 − 0.0480i)13-s + (−1.27 − 0.687i)14-s + (−0.0415 − 0.0505i)15-s − 0.879·16-s + (0.760 + 1.31i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0735i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0735i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.997 - 0.0735i$
Analytic conductor: \(0.503057\)
Root analytic conductor: \(0.709265\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :1/2),\ 0.997 - 0.0735i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.25608 + 0.0462451i\)
\(L(\frac12)\) \(\approx\) \(1.25608 + 0.0462451i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.70 - 0.283i)T \)
7 \( 1 + (2.33 + 1.25i)T \)
good2 \( 1 - 2.05T + 2T^{2} \)
5 \( 1 + (-0.0731 - 0.126i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.832 - 1.44i)T + (-5.5 - 9.52i)T^{2} \)
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 \( 1 + (-3.09 - 5.35i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.46 + 4.27i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 2.51T + 31T^{2} \)
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 \( 1 + 1.81T + 47T^{2} \)
53 \( 1 + (2.67 + 4.62i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 4.57T + 59T^{2} \)
61 \( 1 + 0.678T + 61T^{2} \)
67 \( 1 + 6.18T + 67T^{2} \)
71 \( 1 - 1.27T + 71T^{2} \)
73 \( 1 + (0.778 + 1.34i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 12.7T + 79T^{2} \)
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

−15.06824978159329019458757615968, −13.58431647917980547469410239088, −12.86944437053201546120144885181, −11.97244737586254092887587628728, −10.80325049402749215865202121244, −9.599066149654337213063185160549, −7.13786940945614657503953669295, −6.07013110388134911569095207318, −4.89905415517488434285814136983, −3.51326071174946722430497290653, 3.29039209831810741951829606088, 5.10695350308937058297288025336, 5.91132333595949228298131116926, 7.15705947047609295091261253190, 9.399500640718979908573771597269, 10.89884951958082415301655838415, 12.10384278872832847943992416447, 12.63345945180051057622454307322, 13.67055418467688269682774368009, 14.80465538995882276779622766777

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