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Properties

Label 2-63-9.7-c3-0-11
Degree $2$
Conductor $63$
Sign $0.247 + 0.968i$
Analytic cond. $3.71712$
Root an. cond. $1.92798$
Motivic weight $3$
Arithmetic yes
Rational no
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: $0.247 + 0.968i$
Analytic conductor: \(3.71712\)
Root analytic conductor: \(1.92798\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (43, \cdot )$
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

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

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