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Properties

Label 2-63-63.41-c1-0-4
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: $\chi_{63} (41, \cdot )$
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

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

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