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Properties

Label 2-69-23.22-c4-0-7
Degree $2$
Conductor $69$
Sign $-0.634 + 0.772i$
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  − 5.20·2-s − 5.19·3-s + 11.0·4-s + 30.5i·5-s + 27.0·6-s + 6.50i·7-s + 25.7·8-s + 27·9-s − 158. i·10-s + 127. i·11-s − 57.4·12-s − 221.·13-s − 33.8i·14-s − 158. i·15-s − 310.·16-s − 381. i·17-s + ⋯
L(s)  = 1  − 1.30·2-s − 0.577·3-s + 0.691·4-s + 1.22i·5-s + 0.750·6-s + 0.132i·7-s + 0.401·8-s + 0.333·9-s − 1.58i·10-s + 1.05i·11-s − 0.398·12-s − 1.31·13-s − 0.172i·14-s − 0.705i·15-s − 1.21·16-s − 1.31i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.634 + 0.772i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.634 + 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $-0.634 + 0.772i$
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: \((2,\ 69,\ (\ :2),\ -0.634 + 0.772i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.0168857 - 0.0357220i\)
\(L(\frac12)\) \(\approx\) \(0.0168857 - 0.0357220i\)
\(L(3)\) 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 \( 1 + (335. - 408. i)T \)
good2 \( 1 + 5.20T + 16T^{2} \)
5 \( 1 - 30.5iT - 625T^{2} \)
7 \( 1 - 6.50iT - 2.40e3T^{2} \)
11 \( 1 - 127. iT - 1.46e4T^{2} \)
13 \( 1 + 221.T + 2.85e4T^{2} \)
17 \( 1 + 381. iT - 8.35e4T^{2} \)
19 \( 1 + 573. iT - 1.30e5T^{2} \)
29 \( 1 - 217.T + 7.07e5T^{2} \)
31 \( 1 + 509.T + 9.23e5T^{2} \)
37 \( 1 + 76.8iT - 1.87e6T^{2} \)
41 \( 1 - 620.T + 2.82e6T^{2} \)
43 \( 1 + 3.39e3iT - 3.41e6T^{2} \)
47 \( 1 + 2.38e3T + 4.87e6T^{2} \)
53 \( 1 + 1.77e3iT - 7.89e6T^{2} \)
59 \( 1 + 4.49e3T + 1.21e7T^{2} \)
61 \( 1 - 368. iT - 1.38e7T^{2} \)
67 \( 1 + 2.47e3iT - 2.01e7T^{2} \)
71 \( 1 + 2.68e3T + 2.54e7T^{2} \)
73 \( 1 + 5.50e3T + 2.83e7T^{2} \)
79 \( 1 - 7.44e3iT - 3.89e7T^{2} \)
83 \( 1 + 9.96e3iT - 4.74e7T^{2} \)
89 \( 1 - 1.41e4iT - 6.27e7T^{2} \)
97 \( 1 - 3.34e3iT - 8.85e7T^{2} \)
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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

−13.68317188410879972836371643907, −12.04522296108694089762560690565, −11.05217781597244980086939769324, −10.05152602445145371246007637308, −9.308076315425312276030794031793, −7.36269134579237048458917827162, −7.01006232711321164696625226896, −4.88171319974960862468066447852, −2.37441831184767252832948321946, −0.03645518155126374923622621287, 1.39107846331112789166424572491, 4.46803734737636334317994014122, 5.98477032525045856574857452003, 7.79664577008872720854559944843, 8.597495999682709117558484164650, 9.795125661281140462741574502341, 10.69169679310613749758631508631, 12.08396927045034809030135022678, 12.97100179504635254009139882714, 14.45943075068524231957101380205

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