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Properties

Label 2-63-63.4-c1-0-2
Degree $2$
Conductor $63$
Sign $0.996 - 0.0815i$
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  + (−0.335 + 0.580i)2-s + (−0.377 − 1.69i)3-s + (0.775 + 1.34i)4-s + 1.42·5-s + (1.10 + 0.347i)6-s + (2.21 − 1.44i)7-s − 2.38·8-s + (−2.71 + 1.27i)9-s + (−0.477 + 0.827i)10-s − 4.93·11-s + (1.97 − 1.81i)12-s + (−1.37 + 2.38i)13-s + (0.0972 + 1.77i)14-s + (−0.537 − 2.40i)15-s + (−0.752 + 1.30i)16-s + (0.559 − 0.969i)17-s + ⋯
L(s)  = 1  + (−0.236 + 0.410i)2-s + (−0.217 − 0.975i)3-s + (0.387 + 0.671i)4-s + 0.637·5-s + (0.452 + 0.141i)6-s + (0.837 − 0.546i)7-s − 0.841·8-s + (−0.905 + 0.425i)9-s + (−0.151 + 0.261i)10-s − 1.48·11-s + (0.570 − 0.524i)12-s + (−0.381 + 0.661i)13-s + (0.0259 + 0.473i)14-s + (−0.138 − 0.621i)15-s + (−0.188 + 0.326i)16-s + (0.135 − 0.235i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0815i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0815i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.996 - 0.0815i$
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: \((2,\ 63,\ (\ :1/2),\ 0.996 - 0.0815i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.845532 + 0.0345271i\)
\(L(\frac12)\) \(\approx\) \(0.845532 + 0.0345271i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.377 + 1.69i)T \)
7 \( 1 + (-2.21 + 1.44i)T \)
good2 \( 1 + (0.335 - 0.580i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 - 1.42T + 5T^{2} \)
11 \( 1 + 4.93T + 11T^{2} \)
13 \( 1 + (1.37 - 2.38i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.559 + 0.969i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.00 + 3.47i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 5.43T + 23T^{2} \)
29 \( 1 + (-3.40 - 5.89i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.25 + 2.17i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.709 - 1.22i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.124 + 0.215i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.498 + 0.863i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.73 + 8.20i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.410 - 0.710i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.29 - 5.70i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.0376 - 0.0651i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.29 - 10.9i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 0.0804T + 71T^{2} \)
73 \( 1 + (-5.34 + 9.25i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.922 + 1.59i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.23 + 12.5i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-6.76 - 11.7i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.70 - 4.67i)T + (-48.5 + 84.0i)T^{2} \)
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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

−14.98446795700252637283799713248, −13.69176548065970242328307141540, −12.92015519761970628177184036033, −11.69984246674261101499035914297, −10.67025941076567510576401190927, −8.759029338414712789645250969721, −7.64492662535632723445952402770, −6.83534364245412976352260609106, −5.23491954184880197744602844310, −2.43276306555582225040876355560, 2.55410182050563309062851569300, 5.09911585727316482119125511722, 5.88249907646017099821222604099, 8.179445350993106075371674004931, 9.595433739461642443887281324057, 10.43512906466667548914009454451, 11.18016019411116836439550005621, 12.50901304462440725403087225598, 14.17379468297214502677178870637, 15.15220572324695818656001059970

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