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Properties

Label 2-63-63.4-c1-0-5
Degree $2$
Conductor $63$
Sign $0.0921 + 0.995i$
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  + (0.920 − 1.59i)2-s + (−1.58 − 0.691i)3-s + (−0.695 − 1.20i)4-s + 1.33·5-s + (−2.56 + 1.89i)6-s + (−2.54 + 0.728i)7-s + 1.12·8-s + (2.04 + 2.19i)9-s + (1.22 − 2.12i)10-s + 1.51·11-s + (0.271 + 2.39i)12-s + (−2.58 + 4.48i)13-s + (−1.17 + 4.72i)14-s + (−2.11 − 0.923i)15-s + (2.42 − 4.19i)16-s + (0.774 − 1.34i)17-s + ⋯
L(s)  = 1  + (0.650 − 1.12i)2-s + (−0.916 − 0.399i)3-s + (−0.347 − 0.601i)4-s + 0.596·5-s + (−1.04 + 0.773i)6-s + (−0.961 + 0.275i)7-s + 0.396·8-s + (0.681 + 0.732i)9-s + (0.388 − 0.673i)10-s + 0.456·11-s + (0.0782 + 0.690i)12-s + (−0.717 + 1.24i)13-s + (−0.315 + 1.26i)14-s + (−0.547 − 0.238i)15-s + (0.605 − 1.04i)16-s + (0.187 − 0.325i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0921 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0921 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.0921 + 0.995i$
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: \((2,\ 63,\ (\ :1/2),\ 0.0921 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.730516 - 0.666043i\)
\(L(\frac12)\) \(\approx\) \(0.730516 - 0.666043i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.58 + 0.691i)T \)
7 \( 1 + (2.54 - 0.728i)T \)
good2 \( 1 + (-0.920 + 1.59i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 - 1.33T + 5T^{2} \)
11 \( 1 - 1.51T + 11T^{2} \)
13 \( 1 + (2.58 - 4.48i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.774 + 1.34i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.25 + 2.16i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 7.36T + 23T^{2} \)
29 \( 1 + (0.0309 + 0.0536i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.92 - 3.33i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.281 + 0.487i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.51 + 7.81i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.09 - 8.83i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.75 + 8.24i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.755 + 1.30i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.22 - 7.31i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.61 - 2.80i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.46 + 6.00i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12.3T + 71T^{2} \)
73 \( 1 + (1.37 - 2.38i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.95 + 5.12i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.80 - 4.85i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-0.703 - 1.21i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.09 + 10.5i)T + (-48.5 + 84.0i)T^{2} \)
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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

−14.12317962924384170669992426211, −13.34684560679393061362192899367, −12.22791006887925045519944010536, −11.78013689490419142948636876651, −10.40940358617754387690429970364, −9.486210906963728923912868917412, −7.12015374074903283108232334223, −5.86321895674868584610190628483, −4.28300465938008712829223947699, −2.18316648232411161285618137625, 4.09848401459474723213250081591, 5.70052842987493065134410140643, 6.24951606621559275015337977984, 7.63918736729866026277841160796, 9.759393757349704769318541458103, 10.46964078184173854575996239036, 12.25108632471636500946889406339, 13.16054281025818170942149562487, 14.33719508628848535565203938060, 15.40968017430480391531138062498

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