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Properties

Label 2-69-69.68-c7-0-13
Degree $2$
Conductor $69$
Sign $-0.604 + 0.796i$
Analytic cond. $21.5545$
Root an. cond. $4.64268$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 15.9i·2-s + (37.2 + 28.2i)3-s − 125.·4-s + (−450. + 593. i)6-s + 38.2i·8-s + (589. + 2.10e3i)9-s + (−4.67e3 − 3.54e3i)12-s − 1.39e4·13-s − 1.66e4·16-s + (−3.35e4 + 9.39e3i)18-s + 5.83e4i·23-s + (−1.08e3 + 1.42e3i)24-s − 7.81e4·25-s − 2.21e5i·26-s + (−3.75e4 + 9.51e4i)27-s + ⋯
L(s)  = 1  + 1.40i·2-s + (0.796 + 0.604i)3-s − 0.981·4-s + (−0.850 + 1.12i)6-s + 0.0263i·8-s + (0.269 + 0.962i)9-s + (−0.781 − 0.592i)12-s − 1.75·13-s − 1.01·16-s + (−1.35 + 0.379i)18-s + 0.999i·23-s + (−0.0159 + 0.0210i)24-s − 25-s − 2.47i·26-s + (−0.367 + 0.930i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.604 + 0.796i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.604 + 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $-0.604 + 0.796i$
Analytic conductor: \(21.5545\)
Root analytic conductor: \(4.64268\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{69} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 69,\ (\ :7/2),\ -0.604 + 0.796i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.740533 - 1.49111i\)
\(L(\frac12)\) \(\approx\) \(0.740533 - 1.49111i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-37.2 - 28.2i)T \)
23 \( 1 - 5.83e4iT \)
good2 \( 1 - 15.9iT - 128T^{2} \)
5 \( 1 + 7.81e4T^{2} \)
7 \( 1 - 8.23e5T^{2} \)
11 \( 1 + 1.94e7T^{2} \)
13 \( 1 + 1.39e4T + 6.27e7T^{2} \)
17 \( 1 + 4.10e8T^{2} \)
19 \( 1 - 8.93e8T^{2} \)
29 \( 1 - 2.68e3iT - 1.72e10T^{2} \)
31 \( 1 - 3.25e5T + 2.75e10T^{2} \)
37 \( 1 - 9.49e10T^{2} \)
41 \( 1 + 5.25e5iT - 1.94e11T^{2} \)
43 \( 1 - 2.71e11T^{2} \)
47 \( 1 - 4.87e5iT - 5.06e11T^{2} \)
53 \( 1 + 1.17e12T^{2} \)
59 \( 1 - 3.15e6iT - 2.48e12T^{2} \)
61 \( 1 - 3.14e12T^{2} \)
67 \( 1 - 6.06e12T^{2} \)
71 \( 1 - 3.47e6iT - 9.09e12T^{2} \)
73 \( 1 + 6.00e5T + 1.10e13T^{2} \)
79 \( 1 - 1.92e13T^{2} \)
83 \( 1 + 2.71e13T^{2} \)
89 \( 1 + 4.42e13T^{2} \)
97 \( 1 - 8.07e13T^{2} \)
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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

−14.30040887096031220383065320228, −13.52827247284450762029358904132, −11.86779944025185782026835773989, −10.16740632856589455012956879496, −9.173863986833171649642398184104, −7.962620143188958155602132905188, −7.17726735098294056277776633400, −5.52460753319555626815900683318, −4.38733316780046842222768161479, −2.48511157158735619661437952008, 0.49669506052699704284001331789, 2.03966920362277573817680264111, 2.94238268434458698608580510949, 4.45694803116396329460303177334, 6.72201351025515948888498709232, 8.052872467221170707389818509650, 9.459837270100930676055154327317, 10.19949955713837777259538009142, 11.77368703328419168535876403559, 12.37999570358437345520849895464

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