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Properties

Label 2-69-23.22-c4-0-9
Degree $2$
Conductor $69$
Sign $-0.114 + 0.993i$
Analytic cond. $7.13252$
Root an. cond. $2.67067$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 6.46·2-s + 5.19·3-s + 25.8·4-s − 30.6i·5-s − 33.6·6-s + 37.9i·7-s − 63.5·8-s + 27·9-s + 198. i·10-s − 2.87i·11-s + 134.·12-s − 18.8·13-s − 245. i·14-s − 159. i·15-s − 2.35·16-s − 445. i·17-s + ⋯
L(s)  = 1  − 1.61·2-s + 0.577·3-s + 1.61·4-s − 1.22i·5-s − 0.933·6-s + 0.775i·7-s − 0.992·8-s + 0.333·9-s + 1.98i·10-s − 0.0237i·11-s + 0.931·12-s − 0.111·13-s − 1.25i·14-s − 0.708i·15-s − 0.00920·16-s − 1.54i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.114 + 0.993i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.114 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $-0.114 + 0.993i$
Analytic conductor: \(7.13252\)
Root analytic conductor: \(2.67067\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{69} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 69,\ (\ :2),\ -0.114 + 0.993i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.490028 - 0.549944i\)
\(L(\frac12)\) \(\approx\) \(0.490028 - 0.549944i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 5.19T \)
23 \( 1 + (-60.7 + 525. i)T \)
good2 \( 1 + 6.46T + 16T^{2} \)
5 \( 1 + 30.6iT - 625T^{2} \)
7 \( 1 - 37.9iT - 2.40e3T^{2} \)
11 \( 1 + 2.87iT - 1.46e4T^{2} \)
13 \( 1 + 18.8T + 2.85e4T^{2} \)
17 \( 1 + 445. iT - 8.35e4T^{2} \)
19 \( 1 + 401. iT - 1.30e5T^{2} \)
29 \( 1 + 922.T + 7.07e5T^{2} \)
31 \( 1 - 767.T + 9.23e5T^{2} \)
37 \( 1 + 1.75e3iT - 1.87e6T^{2} \)
41 \( 1 + 2.56e3T + 2.82e6T^{2} \)
43 \( 1 - 17.6iT - 3.41e6T^{2} \)
47 \( 1 + 2.03e3T + 4.87e6T^{2} \)
53 \( 1 - 1.30e3iT - 7.89e6T^{2} \)
59 \( 1 - 4.38e3T + 1.21e7T^{2} \)
61 \( 1 - 735. iT - 1.38e7T^{2} \)
67 \( 1 + 3.48e3iT - 2.01e7T^{2} \)
71 \( 1 - 5.66e3T + 2.54e7T^{2} \)
73 \( 1 - 2.21e3T + 2.83e7T^{2} \)
79 \( 1 - 3.99e3iT - 3.89e7T^{2} \)
83 \( 1 - 9.28e3iT - 4.74e7T^{2} \)
89 \( 1 - 1.09e4iT - 6.27e7T^{2} \)
97 \( 1 + 1.02e4iT - 8.85e7T^{2} \)
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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

−13.62358569951224287386065182994, −12.39119115997744420310644123279, −11.25320633194393132927994364834, −9.686807948886262936855835610069, −9.015432133229164014737641637409, −8.328345997619694008527758086479, −7.02447262166693788028141325565, −4.97310319197061081827952460720, −2.35957438989637068436349708583, −0.62824000869138676055097175890, 1.71418863071570087281945245461, 3.53154235586323410122898331788, 6.52332340872670004242978717721, 7.52933031625548505032964972561, 8.408530666111986694866492447266, 9.955596102906861584854068211502, 10.38089912183531569123926535278, 11.50617033573423709427781143213, 13.33850079827957438157653143696, 14.57122716139181273928535070835

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