Properties

Label 2-69-69.68-c5-0-0
Degree $2$
Conductor $69$
Sign $-0.851 + 0.523i$
Analytic cond. $11.0664$
Root an. cond. $3.32663$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.57i·2-s + (−12.8 + 8.75i)3-s + 29.5·4-s − 37.1·5-s + (−13.7 − 20.2i)6-s + 122. i·7-s + 96.6i·8-s + (89.6 − 225. i)9-s − 58.4i·10-s − 321.·11-s + (−380. + 258. i)12-s − 207.·13-s − 192.·14-s + (479. − 325. i)15-s + 793.·16-s − 1.64e3·17-s + ⋯
L(s)  = 1  + 0.277i·2-s + (−0.827 + 0.561i)3-s + 0.922·4-s − 0.665·5-s + (−0.156 − 0.229i)6-s + 0.942i·7-s + 0.534i·8-s + (0.368 − 0.929i)9-s − 0.184i·10-s − 0.801·11-s + (−0.763 + 0.518i)12-s − 0.340·13-s − 0.261·14-s + (0.550 − 0.373i)15-s + 0.774·16-s − 1.37·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.851 + 0.523i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.851 + 0.523i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $-0.851 + 0.523i$
Analytic conductor: \(11.0664\)
Root analytic conductor: \(3.32663\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{69} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 69,\ (\ :5/2),\ -0.851 + 0.523i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.0821300 - 0.290284i\)
\(L(\frac12)\) \(\approx\) \(0.0821300 - 0.290284i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (12.8 - 8.75i)T \)
23 \( 1 + (2.53e3 + 114. i)T \)
good2 \( 1 - 1.57iT - 32T^{2} \)
5 \( 1 + 37.1T + 3.12e3T^{2} \)
7 \( 1 - 122. iT - 1.68e4T^{2} \)
11 \( 1 + 321.T + 1.61e5T^{2} \)
13 \( 1 + 207.T + 3.71e5T^{2} \)
17 \( 1 + 1.64e3T + 1.41e6T^{2} \)
19 \( 1 + 2.88e3iT - 2.47e6T^{2} \)
29 \( 1 + 2.44e3iT - 2.05e7T^{2} \)
31 \( 1 + 244.T + 2.86e7T^{2} \)
37 \( 1 - 9.93e3iT - 6.93e7T^{2} \)
41 \( 1 - 1.74e4iT - 1.15e8T^{2} \)
43 \( 1 - 3.35e3iT - 1.47e8T^{2} \)
47 \( 1 + 1.50e4iT - 2.29e8T^{2} \)
53 \( 1 - 7.91e3T + 4.18e8T^{2} \)
59 \( 1 + 1.13e4iT - 7.14e8T^{2} \)
61 \( 1 - 2.59e4iT - 8.44e8T^{2} \)
67 \( 1 - 3.69e4iT - 1.35e9T^{2} \)
71 \( 1 - 5.28e4iT - 1.80e9T^{2} \)
73 \( 1 + 7.18e4T + 2.07e9T^{2} \)
79 \( 1 + 4.33e4iT - 3.07e9T^{2} \)
83 \( 1 + 1.07e5T + 3.93e9T^{2} \)
89 \( 1 - 1.12e3T + 5.58e9T^{2} \)
97 \( 1 - 2.67e4iT - 8.58e9T^{2} \)
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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

−15.10981816964075272167238151552, −13.07183357472506690832429800690, −11.68797849100888831471726096601, −11.41510568359146257611059394974, −10.07898140545330651130560359965, −8.550144246290115999739867015437, −7.09073480268786726226498414810, −5.94043935093083061912441610208, −4.65024713490708582208759881902, −2.59514621490958248657909700764, 0.13629943859688969824557755578, 1.93540279306998961845953848103, 4.02092186367666396731034276755, 5.86945294446196772370886188829, 7.17975839128468495658517875558, 7.86744900818822610412261176244, 10.28866204554034685559326446929, 10.89359521886072708207574795428, 11.96942508581739319369776670412, 12.73724175440538526055198391990

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