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Properties

Label 2-69-3.2-c6-0-18
Degree $2$
Conductor $69$
Sign $0.208 - 0.978i$
Analytic cond. $15.8737$
Root an. cond. $3.98418$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 15.2i·2-s + (−26.4 − 5.62i)3-s − 167.·4-s + 4.50i·5-s + (85.5 − 401. i)6-s + 443.·7-s − 1.56e3i·8-s + (665. + 297. i)9-s − 68.4·10-s − 1.98e3i·11-s + (4.41e3 + 940. i)12-s − 3.41e3·13-s + 6.74e3i·14-s + (25.3 − 118. i)15-s + 1.31e4·16-s + 4.05e3i·17-s + ⋯
L(s)  = 1  + 1.90i·2-s + (−0.978 − 0.208i)3-s − 2.61·4-s + 0.0360i·5-s + (0.395 − 1.85i)6-s + 1.29·7-s − 3.06i·8-s + (0.913 + 0.407i)9-s − 0.0684·10-s − 1.49i·11-s + (2.55 + 0.544i)12-s − 1.55·13-s + 2.45i·14-s + (0.00750 − 0.0352i)15-s + 3.21·16-s + 0.825i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.208 - 0.978i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.208 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $0.208 - 0.978i$
Analytic conductor: \(15.8737\)
Root analytic conductor: \(3.98418\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{69} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 69,\ (\ :3),\ 0.208 - 0.978i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.823620 + 0.666668i\)
\(L(\frac12)\) \(\approx\) \(0.823620 + 0.666668i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (26.4 + 5.62i)T \)
23 \( 1 + 2.53e3iT \)
good2 \( 1 - 15.2iT - 64T^{2} \)
5 \( 1 - 4.50iT - 1.56e4T^{2} \)
7 \( 1 - 443.T + 1.17e5T^{2} \)
11 \( 1 + 1.98e3iT - 1.77e6T^{2} \)
13 \( 1 + 3.41e3T + 4.82e6T^{2} \)
17 \( 1 - 4.05e3iT - 2.41e7T^{2} \)
19 \( 1 - 1.55e3T + 4.70e7T^{2} \)
29 \( 1 + 2.94e3iT - 5.94e8T^{2} \)
31 \( 1 - 5.14e4T + 8.87e8T^{2} \)
37 \( 1 - 2.69e4T + 2.56e9T^{2} \)
41 \( 1 - 2.84e4iT - 4.75e9T^{2} \)
43 \( 1 - 4.97e4T + 6.32e9T^{2} \)
47 \( 1 + 1.26e5iT - 1.07e10T^{2} \)
53 \( 1 + 1.56e5iT - 2.21e10T^{2} \)
59 \( 1 + 1.43e4iT - 4.21e10T^{2} \)
61 \( 1 - 2.71e4T + 5.15e10T^{2} \)
67 \( 1 - 8.08e4T + 9.04e10T^{2} \)
71 \( 1 - 5.83e5iT - 1.28e11T^{2} \)
73 \( 1 - 5.94e4T + 1.51e11T^{2} \)
79 \( 1 - 1.28e5T + 2.43e11T^{2} \)
83 \( 1 + 1.06e6iT - 3.26e11T^{2} \)
89 \( 1 + 5.95e5iT - 4.96e11T^{2} \)
97 \( 1 - 1.21e6T + 8.32e11T^{2} \)
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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

−14.15543353436729223338682027640, −12.99567635682780321433735864289, −11.62941190224967099309223037723, −10.19701583313281314976666400232, −8.532237461067447479341454044322, −7.68352721598220947750862373278, −6.49597012840150128834765822931, −5.39806907087534029540889266827, −4.56065929275047006486826369877, −0.65679066795612415766082471650, 1.00846368737762621801378012903, 2.36690937734265361248699724651, 4.64309555383169634542396063525, 4.86338143670471688062928508589, 7.54256222662041236630340748276, 9.387572171906062077636256381407, 10.16508488218218053290147620190, 11.17946155642148065471313476094, 12.06485521641288543839645091612, 12.53119096419385249405367643108

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