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Properties

Label 2-63-63.47-c1-0-3
Degree $2$
Conductor $63$
Sign $0.935 + 0.352i$
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  + (1.81 − 1.04i)2-s + (−0.769 + 1.55i)3-s + (1.19 − 2.07i)4-s − 2.08·5-s + (0.228 + 3.62i)6-s + (−0.879 − 2.49i)7-s − 0.819i·8-s + (−1.81 − 2.38i)9-s + (−3.79 + 2.18i)10-s + 3.22i·11-s + (2.29 + 3.44i)12-s + (2.68 − 1.55i)13-s + (−4.21 − 3.60i)14-s + (1.60 − 3.24i)15-s + (1.53 + 2.65i)16-s + (−0.816 − 1.41i)17-s + ⋯
L(s)  = 1  + (1.28 − 0.740i)2-s + (−0.444 + 0.895i)3-s + (0.597 − 1.03i)4-s − 0.934·5-s + (0.0933 + 1.47i)6-s + (−0.332 − 0.943i)7-s − 0.289i·8-s + (−0.604 − 0.796i)9-s + (−1.19 + 0.692i)10-s + 0.973i·11-s + (0.661 + 0.995i)12-s + (0.745 − 0.430i)13-s + (−1.12 − 0.964i)14-s + (0.415 − 0.837i)15-s + (0.383 + 0.663i)16-s + (−0.197 − 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.935 + 0.352i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.935 + 0.352i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.935 + 0.352i$
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: \((2,\ 63,\ (\ :1/2),\ 0.935 + 0.352i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.22705 - 0.223329i\)
\(L(\frac12)\) \(\approx\) \(1.22705 - 0.223329i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.769 - 1.55i)T \)
7 \( 1 + (0.879 + 2.49i)T \)
good2 \( 1 + (-1.81 + 1.04i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + 2.08T + 5T^{2} \)
11 \( 1 - 3.22iT - 11T^{2} \)
13 \( 1 + (-2.68 + 1.55i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.816 + 1.41i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.79 - 2.76i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 1.16iT - 23T^{2} \)
29 \( 1 + (7.05 + 4.07i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (5.16 + 2.98i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.82 + 4.89i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.35 + 2.34i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.974 - 1.68i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.06 - 7.04i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.27 + 3.04i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.98 - 3.43i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.15 - 2.39i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.336 + 0.583i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 7.01iT - 71T^{2} \)
73 \( 1 + (2.96 - 1.71i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.07 - 12.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.54 + 2.67i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (2.45 - 4.25i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.07 - 1.20i)T + (48.5 + 84.0i)T^{2} \)
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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

−14.88420719568037966651222679634, −13.72144101588468943964414662700, −12.58230220492884632558623328157, −11.56899406191936084100890231245, −10.83679474515010501120371074837, −9.639929069181980288635582641867, −7.57736648942104953964436731259, −5.69232466547016936706698071670, −4.25949135515091530670074020028, −3.56680784594110894288333789375, 3.42728543789873947427918259141, 5.30410482081078681096183615632, 6.28517269882722979852269382607, 7.43238040723124381417570351655, 8.748481767118453471241830650215, 11.25616382739969316946305204393, 11.98082331526197747002132744080, 13.01099160241084058480711067148, 13.79982605376034818925904877134, 15.02446582258469027358430994229

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