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Properties

Label 2-63-63.38-c1-0-2
Degree $2$
Conductor $63$
Sign $0.762 - 0.647i$
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  + 1.51i·2-s + (0.811 − 1.53i)3-s − 0.280·4-s + (−0.387 + 0.671i)5-s + (2.31 + 1.22i)6-s + (−2.46 − 0.952i)7-s + 2.59i·8-s + (−1.68 − 2.48i)9-s + (−1.01 − 0.585i)10-s + (−3.32 + 1.92i)11-s + (−0.227 + 0.429i)12-s + (2.54 − 1.46i)13-s + (1.43 − 3.72i)14-s + (0.713 + 1.13i)15-s − 4.48·16-s + (2.69 − 4.67i)17-s + ⋯
L(s)  = 1  + 1.06i·2-s + (0.468 − 0.883i)3-s − 0.140·4-s + (−0.173 + 0.300i)5-s + (0.943 + 0.500i)6-s + (−0.933 − 0.359i)7-s + 0.918i·8-s + (−0.561 − 0.827i)9-s + (−0.320 − 0.185i)10-s + (−1.00 + 0.579i)11-s + (−0.0656 + 0.123i)12-s + (0.705 − 0.407i)13-s + (0.384 − 0.996i)14-s + (0.184 + 0.294i)15-s − 1.12·16-s + (0.654 − 1.13i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.762 - 0.647i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.762 - 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.762 - 0.647i$
Analytic conductor: \(0.503057\)
Root analytic conductor: \(0.709265\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :1/2),\ 0.762 - 0.647i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.910397 + 0.334290i\)
\(L(\frac12)\) \(\approx\) \(0.910397 + 0.334290i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.811 + 1.53i)T \)
7 \( 1 + (2.46 + 0.952i)T \)
good2 \( 1 - 1.51iT - 2T^{2} \)
5 \( 1 + (0.387 - 0.671i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (3.32 - 1.92i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.54 + 1.46i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.69 + 4.67i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.376 - 0.217i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.0482 + 0.0278i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.187 + 0.108i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 6.55iT - 31T^{2} \)
37 \( 1 + (-3.14 - 5.45i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.78 + 6.56i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-6.42 + 11.1i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 0.965T + 47T^{2} \)
53 \( 1 + (-6.46 - 3.73i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 3.12T + 59T^{2} \)
61 \( 1 - 3.48iT - 61T^{2} \)
67 \( 1 + 4.20T + 67T^{2} \)
71 \( 1 - 3.50iT - 71T^{2} \)
73 \( 1 + (7.05 + 4.07i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + 4.96T + 79T^{2} \)
83 \( 1 + (-4.31 + 7.46i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (7.82 + 13.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.24 + 0.716i)T + (48.5 + 84.0i)T^{2} \)
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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

−15.24672409881902345362950249872, −14.06060840535854398012590319877, −13.22031318561206211265869557158, −12.02832024409503267625322218641, −10.48475097994109060744342268715, −8.852992669753384668812670949987, −7.52746124030261691536421550696, −6.96829865227808338489731227033, −5.61673001116118223376384344322, −2.96695900681346403200622402915, 2.78042124615676829566658450800, 4.00649930140682397576822925612, 6.00390087805216955030305663697, 8.174878474850850195173319652083, 9.415631288584645088849820255659, 10.37156703435585623252520414557, 11.25517199115975028988471671606, 12.62465379583745425306444353662, 13.42583500996887467056344415306, 14.98619406027910004368724781316

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