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Properties

Label 2-63-63.4-c1-0-3
Degree $2$
Conductor $63$
Sign $0.952 + 0.305i$
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  + (0.247 − 0.429i)2-s + (1.59 − 0.667i)3-s + (0.877 + 1.51i)4-s − 3.69·5-s + (0.109 − 0.851i)6-s + (−2.60 − 0.436i)7-s + 1.86·8-s + (2.10 − 2.13i)9-s + (−0.915 + 1.58i)10-s − 0.892·11-s + (2.41 + 1.84i)12-s + (0.598 − 1.03i)13-s + (−0.834 + 1.01i)14-s + (−5.90 + 2.46i)15-s + (−1.29 + 2.23i)16-s + (−0.124 + 0.216i)17-s + ⋯
L(s)  = 1  + (0.175 − 0.303i)2-s + (0.922 − 0.385i)3-s + (0.438 + 0.759i)4-s − 1.65·5-s + (0.0447 − 0.347i)6-s + (−0.986 − 0.165i)7-s + 0.658·8-s + (0.703 − 0.711i)9-s + (−0.289 + 0.501i)10-s − 0.269·11-s + (0.697 + 0.531i)12-s + (0.165 − 0.287i)13-s + (−0.223 + 0.270i)14-s + (−1.52 + 0.636i)15-s + (−0.323 + 0.559i)16-s + (−0.0303 + 0.0525i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 + 0.305i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.952 + 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.952 + 0.305i$
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: \((2,\ 63,\ (\ :1/2),\ 0.952 + 0.305i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03543 - 0.162240i\)
\(L(\frac12)\) \(\approx\) \(1.03543 - 0.162240i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.59 + 0.667i)T \)
7 \( 1 + (2.60 + 0.436i)T \)
good2 \( 1 + (-0.247 + 0.429i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + 3.69T + 5T^{2} \)
11 \( 1 + 0.892T + 11T^{2} \)
13 \( 1 + (-0.598 + 1.03i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.124 - 0.216i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.40 - 2.43i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 2.47T + 23T^{2} \)
29 \( 1 + (-2.07 - 3.58i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.79 + 3.10i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.36 + 4.09i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.39 - 4.14i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.98 + 8.64i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.08 + 8.81i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.94 - 8.56i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.906 + 1.56i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.40 - 9.35i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.514 + 0.891i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 4.94T + 71T^{2} \)
73 \( 1 + (0.915 - 1.58i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.899 + 1.55i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6.16 - 10.6i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.20 + 2.08i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.52 - 9.56i)T + (-48.5 + 84.0i)T^{2} \)
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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

−15.13278466354339477210126506384, −13.55943510190503932811052544512, −12.58868957808806882832052972310, −11.94500405629316739096696827882, −10.54863121386110223888548405422, −8.771645737371555928118105047042, −7.72962561025683482540443523625, −6.97062221906745207652555839037, −3.93268315212982890138218541249, −3.10381783861810109463998122876, 3.18472969879382765145417708865, 4.69361569029221378036544158127, 6.76207320829782240910166171102, 7.82626774341330363998532330537, 9.181859333322193367524753569343, 10.45038206493668952876901249079, 11.59538544729436281647533088022, 13.00143081791790352057865498478, 14.23110221078769250069484115764, 15.36374647682216774783981975161

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