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Properties

Label 2-63-63.32-c2-0-6
Degree $2$
Conductor $63$
Sign $0.494 - 0.869i$
Analytic cond. $1.71662$
Root an. cond. $1.31020$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.48 + 1.43i)2-s + (0.950 + 2.84i)3-s + (2.12 + 3.67i)4-s − 7.54i·5-s + (−1.72 + 8.44i)6-s + (−6.82 − 1.56i)7-s + 0.709i·8-s + (−7.19 + 5.40i)9-s + (10.8 − 18.7i)10-s + 15.1i·11-s + (−8.44 + 9.53i)12-s + (4.30 − 7.45i)13-s + (−14.7 − 13.6i)14-s + (21.4 − 7.17i)15-s + (7.47 − 12.9i)16-s + (4.60 + 2.66i)17-s + ⋯
L(s)  = 1  + (1.24 + 0.717i)2-s + (0.316 + 0.948i)3-s + (0.530 + 0.919i)4-s − 1.50i·5-s + (−0.286 + 1.40i)6-s + (−0.974 − 0.223i)7-s + 0.0886i·8-s + (−0.799 + 0.601i)9-s + (1.08 − 1.87i)10-s + 1.38i·11-s + (−0.703 + 0.794i)12-s + (0.331 − 0.573i)13-s + (−1.05 − 0.977i)14-s + (1.43 − 0.478i)15-s + (0.467 − 0.809i)16-s + (0.271 + 0.156i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.494 - 0.869i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.494 - 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.494 - 0.869i$
Analytic conductor: \(1.71662\)
Root analytic conductor: \(1.31020\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :1),\ 0.494 - 0.869i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.82663 + 1.06295i\)
\(L(\frac12)\) \(\approx\) \(1.82663 + 1.06295i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.950 - 2.84i)T \)
7 \( 1 + (6.82 + 1.56i)T \)
good2 \( 1 + (-2.48 - 1.43i)T + (2 + 3.46i)T^{2} \)
5 \( 1 + 7.54iT - 25T^{2} \)
11 \( 1 - 15.1iT - 121T^{2} \)
13 \( 1 + (-4.30 + 7.45i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + (-4.60 - 2.66i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (0.417 + 0.722i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 - 39.1iT - 529T^{2} \)
29 \( 1 + (-12.5 + 7.25i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (6.37 + 11.0i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (11.7 + 20.3i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (13.4 + 7.75i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-0.448 - 0.776i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-2.03 - 1.17i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-31.4 - 18.1i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (42.0 - 24.3i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-14.8 + 25.6i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (46.2 + 80.1i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 15.1iT - 5.04e3T^{2} \)
73 \( 1 + (46.8 - 81.0i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-41.0 + 71.1i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-127. + 73.3i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + (92.3 - 53.2i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (26.0 + 45.0i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−15.16943537715102550090918396148, −13.74418094794574870620210828745, −12.98708334148760121568267666405, −12.12511906149774654001767356770, −10.03988176706546889358216872895, −9.191193770109440531927933215480, −7.58774836341992964721761741871, −5.75770076201286592782151853949, −4.76522997410455483397353895260, −3.68270880027893201905190287083, 2.67191756372143186308457536350, 3.44546376207259184882113503691, 6.02093637720244038448257463154, 6.74890794160786039049257298805, 8.565986117779886862670182863697, 10.49812460202619963683443050501, 11.46087069601006340791549200564, 12.43218395827523670042314299599, 13.56328320296168281311483754206, 14.10763100488710467018599807439

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