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Properties

Label 2-69-69.68-c5-0-26
Degree $2$
Conductor $69$
Sign $0.703 + 0.711i$
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  + 2.78i·2-s + (−1.49 − 15.5i)3-s + 24.2·4-s + 76.0·5-s + (43.1 − 4.15i)6-s − 165. i·7-s + 156. i·8-s + (−238. + 46.3i)9-s + 211. i·10-s − 574.·11-s + (−36.2 − 376. i)12-s + 1.11e3·13-s + 461.·14-s + (−113. − 1.17e3i)15-s + 341.·16-s − 321.·17-s + ⋯
L(s)  = 1  + 0.491i·2-s + (−0.0958 − 0.995i)3-s + 0.758·4-s + 1.36·5-s + (0.489 − 0.0471i)6-s − 1.28i·7-s + 0.864i·8-s + (−0.981 + 0.190i)9-s + 0.668i·10-s − 1.43·11-s + (−0.0726 − 0.754i)12-s + 1.82·13-s + 0.629·14-s + (−0.130 − 1.35i)15-s + 0.333·16-s − 0.269·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.703 + 0.711i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.703 + 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $0.703 + 0.711i$
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: \((2,\ 69,\ (\ :5/2),\ 0.703 + 0.711i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.22952 - 0.930729i\)
\(L(\frac12)\) \(\approx\) \(2.22952 - 0.930729i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.49 + 15.5i)T \)
23 \( 1 + (-1.96e3 + 1.60e3i)T \)
good2 \( 1 - 2.78iT - 32T^{2} \)
5 \( 1 - 76.0T + 3.12e3T^{2} \)
7 \( 1 + 165. iT - 1.68e4T^{2} \)
11 \( 1 + 574.T + 1.61e5T^{2} \)
13 \( 1 - 1.11e3T + 3.71e5T^{2} \)
17 \( 1 + 321.T + 1.41e6T^{2} \)
19 \( 1 + 1.72e3iT - 2.47e6T^{2} \)
29 \( 1 - 3.17e3iT - 2.05e7T^{2} \)
31 \( 1 + 2.46e3T + 2.86e7T^{2} \)
37 \( 1 + 9.38e3iT - 6.93e7T^{2} \)
41 \( 1 - 9.35e3iT - 1.15e8T^{2} \)
43 \( 1 - 7.96e3iT - 1.47e8T^{2} \)
47 \( 1 + 3.85e3iT - 2.29e8T^{2} \)
53 \( 1 - 3.64e3T + 4.18e8T^{2} \)
59 \( 1 - 1.15e4iT - 7.14e8T^{2} \)
61 \( 1 - 4.84e4iT - 8.44e8T^{2} \)
67 \( 1 - 7.84e3iT - 1.35e9T^{2} \)
71 \( 1 - 4.08e4iT - 1.80e9T^{2} \)
73 \( 1 - 1.51e4T + 2.07e9T^{2} \)
79 \( 1 + 1.84e4iT - 3.07e9T^{2} \)
83 \( 1 + 9.40e4T + 3.93e9T^{2} \)
89 \( 1 + 3.24e4T + 5.58e9T^{2} \)
97 \( 1 - 1.74e5iT - 8.58e9T^{2} \)
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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

−13.35326137956924042538263552412, −13.13881952770442483240247505130, −11.07200348693478337288086657868, −10.59751310764344672272867276104, −8.648505711859440318046883279104, −7.36343911490690255119638871714, −6.47934931521162833824919362808, −5.46222039653916576606927299606, −2.64601525554396233022878977813, −1.19465466204854101721544124039, 1.97474706146993899918546337957, 3.20890836096475706924469961835, 5.50874606689949971314811577419, 6.08773874533369343438481724266, 8.447194218423621491049807153976, 9.621158278980956842073832415763, 10.52591448286951877383755748986, 11.35867526107564396737874477189, 12.72242652844670340813351480734, 13.79586148089551555888478720176

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