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Properties

Label 2-63-21.5-c1-0-0
Degree $2$
Conductor $63$
Sign $0.879 - 0.475i$
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  + (1.22 + 0.707i)2-s + (−1.22 + 2.12i)5-s + (−0.5 − 2.59i)7-s − 2.82i·8-s + (−3 + 1.73i)10-s + (−1.22 + 0.707i)11-s + 5.19i·13-s + (1.22 − 3.53i)14-s + (2.00 − 3.46i)16-s + (−2.44 − 4.24i)17-s + (1.5 + 0.866i)19-s − 2·22-s + (4.89 + 2.82i)23-s + (−0.499 − 0.866i)25-s + (−3.67 + 6.36i)26-s + ⋯
L(s)  = 1  + (0.866 + 0.499i)2-s + (−0.547 + 0.948i)5-s + (−0.188 − 0.981i)7-s − 0.999i·8-s + (−0.948 + 0.547i)10-s + (−0.369 + 0.213i)11-s + 1.44i·13-s + (0.327 − 0.944i)14-s + (0.500 − 0.866i)16-s + (−0.594 − 1.02i)17-s + (0.344 + 0.198i)19-s − 0.426·22-s + (1.02 + 0.589i)23-s + (−0.0999 − 0.173i)25-s + (−0.720 + 1.24i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.879 - 0.475i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.879 - 0.475i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.879 - 0.475i$
Analytic conductor: \(0.503057\)
Root analytic conductor: \(0.709265\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :1/2),\ 0.879 - 0.475i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11601 + 0.282049i\)
\(L(\frac12)\) \(\approx\) \(1.11601 + 0.282049i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (0.5 + 2.59i)T \)
good2 \( 1 + (-1.22 - 0.707i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (1.22 - 2.12i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.22 - 0.707i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 5.19iT - 13T^{2} \)
17 \( 1 + (2.44 + 4.24i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.5 - 0.866i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.89 - 2.82i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.82iT - 29T^{2} \)
31 \( 1 + (-1.5 + 0.866i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 7.34T + 41T^{2} \)
43 \( 1 + T + 43T^{2} \)
47 \( 1 + (-6.12 + 10.6i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.44 + 1.41i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.44 + 4.24i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3 + 1.73i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.5 + 9.52i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 7.07iT - 71T^{2} \)
73 \( 1 + (-1.5 + 0.866i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.5 - 4.33i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 7.34T + 83T^{2} \)
89 \( 1 + (-2.44 + 4.24i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 10.3iT - 97T^{2} \)
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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

−14.93790267340926913152534277200, −13.98119638144625823623265416357, −13.35051574736968918206794066769, −11.76587816788669871789519230054, −10.65669465251123000905145826379, −9.411742933182240567692043448061, −7.23917560581803928233065140131, −6.76390461462736867402788506386, −4.86023199039674832031648035943, −3.56071807803503707916438360535, 2.95784791817886436634704789714, 4.62308331320787136533189059178, 5.72977333252804673059989193190, 8.088397365210213726868367655980, 8.823945871600427473843927325438, 10.70163208774571586404402313269, 11.96715861231290917531681451099, 12.70145166144780351495524342417, 13.31884326004089829084932230672, 14.94963814305440677683374833446

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