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Properties

Label 2-69-23.22-c6-0-0
Degree $2$
Conductor $69$
Sign $-0.220 - 0.975i$
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  − 4.73·2-s − 15.5·3-s − 41.5·4-s + 38.4i·5-s + 73.8·6-s − 655. i·7-s + 499.·8-s + 243·9-s − 181. i·10-s − 589. i·11-s + 648.·12-s − 3.17e3·13-s + 3.10e3i·14-s − 598. i·15-s + 292.·16-s − 909. i·17-s + ⋯
L(s)  = 1  − 0.591·2-s − 0.577·3-s − 0.649·4-s + 0.307i·5-s + 0.341·6-s − 1.91i·7-s + 0.976·8-s + 0.333·9-s − 0.181i·10-s − 0.443i·11-s + 0.375·12-s − 1.44·13-s + 1.13i·14-s − 0.177i·15-s + 0.0714·16-s − 0.185i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.220 - 0.975i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.220 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $-0.220 - 0.975i$
Analytic conductor: \(15.8737\)
Root analytic conductor: \(3.98418\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{69} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 69,\ (\ :3),\ -0.220 - 0.975i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.148723 + 0.186007i\)
\(L(\frac12)\) \(\approx\) \(0.148723 + 0.186007i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 15.5T \)
23 \( 1 + (-2.67e3 - 1.18e4i)T \)
good2 \( 1 + 4.73T + 64T^{2} \)
5 \( 1 - 38.4iT - 1.56e4T^{2} \)
7 \( 1 + 655. iT - 1.17e5T^{2} \)
11 \( 1 + 589. iT - 1.77e6T^{2} \)
13 \( 1 + 3.17e3T + 4.82e6T^{2} \)
17 \( 1 + 909. iT - 2.41e7T^{2} \)
19 \( 1 - 3.91e3iT - 4.70e7T^{2} \)
29 \( 1 + 3.80e4T + 5.94e8T^{2} \)
31 \( 1 + 1.25e4T + 8.87e8T^{2} \)
37 \( 1 - 8.08e4iT - 2.56e9T^{2} \)
41 \( 1 - 3.52e4T + 4.75e9T^{2} \)
43 \( 1 - 4.68e4iT - 6.32e9T^{2} \)
47 \( 1 - 1.47e5T + 1.07e10T^{2} \)
53 \( 1 + 2.54e5iT - 2.21e10T^{2} \)
59 \( 1 + 9.88e4T + 4.21e10T^{2} \)
61 \( 1 - 1.49e5iT - 5.15e10T^{2} \)
67 \( 1 - 3.31e5iT - 9.04e10T^{2} \)
71 \( 1 - 3.71e5T + 1.28e11T^{2} \)
73 \( 1 - 3.15e4T + 1.51e11T^{2} \)
79 \( 1 + 2.42e5iT - 2.43e11T^{2} \)
83 \( 1 - 8.27e5iT - 3.26e11T^{2} \)
89 \( 1 - 1.03e6iT - 4.96e11T^{2} \)
97 \( 1 + 8.33e5iT - 8.32e11T^{2} \)
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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

−13.77705698605662974370335452903, −12.87489590066897143990183930073, −11.24418673128448212404782993366, −10.32706687049096954561706139475, −9.565982457851374025588727619389, −7.77660758720427465643105513815, −7.03672649373912780979686151127, −5.07227962033320584734056992980, −3.82636189939286850597196419463, −1.06720967330281977145668569453, 0.15360866502373244583616448519, 2.21120753061251369756430397352, 4.70825964761459688078761969196, 5.61570546039817193146253313351, 7.41911393525118446287268887263, 8.911682580819947546800920127475, 9.422608924071205034528753960477, 10.85387696571906457525639127091, 12.36566720730436101004537114952, 12.67458773754786589525949870589

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