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Properties

Label 2-63-63.20-c1-0-3
Degree $2$
Conductor $63$
Sign $0.997 + 0.0728i$
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 + 0.592i)2-s + (−0.410 − 1.68i)3-s + (−0.296 − 0.514i)4-s + (1.41 + 2.45i)5-s + (0.576 − 1.97i)6-s + (−2.07 + 1.64i)7-s − 3.07i·8-s + (−2.66 + 1.38i)9-s + 3.36i·10-s + (0.136 + 0.0789i)11-s + (−0.743 + 0.710i)12-s + (−3.41 + 1.97i)13-s + (−3.10 + 0.460i)14-s + (3.55 − 3.39i)15-s + (1.23 − 2.13i)16-s + 4.14·17-s + ⋯
L(s)  = 1  + (0.726 + 0.419i)2-s + (−0.236 − 0.971i)3-s + (−0.148 − 0.257i)4-s + (0.634 + 1.09i)5-s + (0.235 − 0.804i)6-s + (−0.783 + 0.621i)7-s − 1.08i·8-s + (−0.887 + 0.460i)9-s + 1.06i·10-s + (0.0412 + 0.0237i)11-s + (−0.214 + 0.205i)12-s + (−0.947 + 0.546i)13-s + (−0.829 + 0.122i)14-s + (0.917 − 0.876i)15-s + (0.307 − 0.532i)16-s + 1.00·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0728i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0728i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.997 + 0.0728i$
Analytic conductor: \(0.503057\)
Root analytic conductor: \(0.709265\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :1/2),\ 0.997 + 0.0728i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09288 - 0.0398731i\)
\(L(\frac12)\) \(\approx\) \(1.09288 - 0.0398731i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.410 + 1.68i)T \)
7 \( 1 + (2.07 - 1.64i)T \)
good2 \( 1 + (-1.02 - 0.592i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (-1.41 - 2.45i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.136 - 0.0789i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.41 - 1.97i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 4.14T + 17T^{2} \)
19 \( 1 + 6.33iT - 19T^{2} \)
23 \( 1 + (-0.472 + 0.273i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.02 - 2.32i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.112 + 0.0647i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 2.46T + 37T^{2} \)
41 \( 1 + (1.99 + 3.45i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.28 + 5.68i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.33 - 7.50i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 2.60iT - 53T^{2} \)
59 \( 1 + (-1.80 - 3.12i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.91 + 1.68i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.663 + 1.14i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 0.409iT - 71T^{2} \)
73 \( 1 - 15.0iT - 73T^{2} \)
79 \( 1 + (2.16 - 3.74i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.22 + 5.58i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 5.05T + 89T^{2} \)
97 \( 1 + (2.18 + 1.26i)T + (48.5 + 84.0i)T^{2} \)
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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

−14.58869342745717551328750960794, −13.99358312650157875472447206306, −12.97130394614294374719692004034, −11.99338317374461856598354734473, −10.43159995287247631637419447172, −9.263077826012223187592253997269, −7.09951252660746841164728508633, −6.44934755103778060246230638061, −5.31485891018619403983242031454, −2.76770702299374616985877090255, 3.36961447092830018315617165010, 4.71770551818954245430073811110, 5.74668252805204241413601620240, 8.131693364794942510938953344034, 9.529772336242170263621741647222, 10.30903591496237822186212224667, 11.99810382046501042805009967370, 12.68908584308767475004298140356, 13.77174885909853039416678779204, 14.75985467970706474054513029480

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