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Properties

Label 2-63-63.47-c1-0-1
Degree $2$
Conductor $63$
Sign $0.520 - 0.854i$
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  + (−0.621 + 0.359i)2-s + (1.34 + 1.09i)3-s + (−0.742 + 1.28i)4-s − 1.44·5-s + (−1.22 − 0.194i)6-s + (2.19 − 1.47i)7-s − 2.50i·8-s + (0.621 + 2.93i)9-s + (0.900 − 0.519i)10-s − 1.80i·11-s + (−2.40 + 0.920i)12-s + (1.88 − 1.09i)13-s + (−0.834 + 1.70i)14-s + (−1.94 − 1.57i)15-s + (−0.585 − 1.01i)16-s + (−1.95 − 3.38i)17-s + ⋯
L(s)  = 1  + (−0.439 + 0.253i)2-s + (0.776 + 0.629i)3-s + (−0.371 + 0.642i)4-s − 0.647·5-s + (−0.501 − 0.0795i)6-s + (0.829 − 0.558i)7-s − 0.884i·8-s + (0.207 + 0.978i)9-s + (0.284 − 0.164i)10-s − 0.542i·11-s + (−0.692 + 0.265i)12-s + (0.523 − 0.302i)13-s + (−0.222 + 0.456i)14-s + (−0.502 − 0.407i)15-s + (−0.146 − 0.253i)16-s + (−0.473 − 0.820i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.520 - 0.854i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.520 - 0.854i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.520 - 0.854i$
Analytic conductor: \(0.503057\)
Root analytic conductor: \(0.709265\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :1/2),\ 0.520 - 0.854i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.704842 + 0.396060i\)
\(L(\frac12)\) \(\approx\) \(0.704842 + 0.396060i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.34 - 1.09i)T \)
7 \( 1 + (-2.19 + 1.47i)T \)
good2 \( 1 + (0.621 - 0.359i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + 1.44T + 5T^{2} \)
11 \( 1 + 1.80iT - 11T^{2} \)
13 \( 1 + (-1.88 + 1.09i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.95 + 3.38i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.47 + 2.00i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 5.67iT - 23T^{2} \)
29 \( 1 + (-8.49 - 4.90i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.45 + 1.41i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.411 - 0.713i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.90 + 10.2i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.76 - 6.52i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.16 - 2.02i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.996 - 0.575i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.89 - 8.47i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.03 + 1.17i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.156 + 0.270i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.94iT - 71T^{2} \)
73 \( 1 + (-2.42 + 1.40i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.21 + 10.7i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.60 - 6.25i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-5.28 + 9.16i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-13.4 - 7.75i)T + (48.5 + 84.0i)T^{2} \)
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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

−15.42140610488695041917742675044, −14.03135551973567805404771413031, −13.28338812780424662073321509299, −11.63122381983668269020213353953, −10.49423582592837542268508633914, −9.013375013956807899987351180867, −8.234667313546984583706291542971, −7.30005002049924687350982837309, −4.65949997762408433700416045833, −3.47025791933994983015690697732, 1.94896097473872960324454837157, 4.39620690277133779342359245136, 6.36771931966596827811735302805, 8.198255321002033289262003566636, 8.631849627246542559221982050580, 10.12165426120298764308686280735, 11.44221127078706527826462215069, 12.52802469081386763460749163531, 13.87497882128446362487997737743, 14.80128832675033162420699048127

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