Properties

Degree $2$
Conductor $63$
Sign $0.247 - 0.968i$
Motivic weight $3$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.28 − 3.95i)2-s + (1.09 − 5.07i)3-s + (−6.41 + 11.1i)4-s + (−10.3 + 18.0i)5-s + (−22.5 + 7.26i)6-s + (3.5 + 6.06i)7-s + 22.0·8-s + (−24.6 − 11.1i)9-s + 94.8·10-s + (−22.6 − 39.3i)11-s + (49.4 + 44.7i)12-s + (−14.6 + 25.3i)13-s + (15.9 − 27.6i)14-s + (80.0 + 72.5i)15-s + (1.04 + 1.81i)16-s − 98.0·17-s + ⋯
L(s)  = 1  + (−0.806 − 1.39i)2-s + (0.210 − 0.977i)3-s + (−0.801 + 1.38i)4-s + (−0.929 + 1.61i)5-s + (−1.53 + 0.494i)6-s + (0.188 + 0.327i)7-s + 0.973·8-s + (−0.911 − 0.411i)9-s + 3.00·10-s + (−0.622 − 1.07i)11-s + (1.18 + 1.07i)12-s + (−0.312 + 0.541i)13-s + (0.304 − 0.528i)14-s + (1.37 + 1.24i)15-s + (0.0163 + 0.0283i)16-s − 1.39·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.247 - 0.968i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.247 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.247 - 0.968i$
Motivic weight: \(3\)
Character: $\chi_{63} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ 0.247 - 0.968i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0326202 + 0.0253439i\)
\(L(\frac12)\) \(\approx\) \(0.0326202 + 0.0253439i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.09 + 5.07i)T \)
7 \( 1 + (-3.5 - 6.06i)T \)
good2 \( 1 + (2.28 + 3.95i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 + (10.3 - 18.0i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (22.6 + 39.3i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (14.6 - 25.3i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 98.0T + 4.91e3T^{2} \)
19 \( 1 - 31.1T + 6.85e3T^{2} \)
23 \( 1 + (4.19 - 7.26i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-36.1 - 62.5i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-15.4 + 26.7i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 196.T + 5.06e4T^{2} \)
41 \( 1 + (106. - 184. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (118. + 205. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-110. - 190. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 55.9T + 1.48e5T^{2} \)
59 \( 1 + (-327. + 566. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (174. + 302. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (105. - 182. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 548.T + 3.57e5T^{2} \)
73 \( 1 + 266.T + 3.89e5T^{2} \)
79 \( 1 + (-134. - 233. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (312. + 541. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 1.60e3T + 7.04e5T^{2} \)
97 \( 1 + (-145. - 252. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−14.45434406297241202947794698724, −13.33006079599737194658561595262, −11.91240465363029799676467506767, −11.35344242304496623935955900194, −10.55105511801974257947193962600, −8.846125599458267051152651968008, −7.83859784646061725088820906115, −6.56955948576702311529542622094, −3.36674174827280871276542831336, −2.35594366059793525635554377309, 0.03421748582433319819363761749, 4.44951816423763357081177211873, 5.27021829743510409990654143921, 7.40601803430138096429653836349, 8.360309188822309685110774724655, 9.115520000011326954731920680506, 10.26610618556575007032856141003, 11.95820712659091024079126519668, 13.38316694506259779209068540398, 15.00829043396353196609945938300

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