Properties

Label 2-69-1.1-c5-0-13
Degree $2$
Conductor $69$
Sign $1$
Analytic cond. $11.0664$
Root an. cond. $3.32663$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 9.34·2-s + 9·3-s + 55.4·4-s + 70.5·5-s + 84.1·6-s − 89.7·7-s + 218.·8-s + 81·9-s + 659.·10-s − 436.·11-s + 498.·12-s − 258.·13-s − 838.·14-s + 634.·15-s + 272.·16-s + 591.·17-s + 757.·18-s + 1.66e3·19-s + 3.90e3·20-s − 807.·21-s − 4.08e3·22-s − 529·23-s + 1.96e3·24-s + 1.84e3·25-s − 2.42e3·26-s + 729·27-s − 4.97e3·28-s + ⋯
L(s)  = 1  + 1.65·2-s + 0.577·3-s + 1.73·4-s + 1.26·5-s + 0.954·6-s − 0.692·7-s + 1.20·8-s + 0.333·9-s + 2.08·10-s − 1.08·11-s + 0.999·12-s − 0.424·13-s − 1.14·14-s + 0.728·15-s + 0.266·16-s + 0.496·17-s + 0.550·18-s + 1.05·19-s + 2.18·20-s − 0.399·21-s − 1.79·22-s − 0.208·23-s + 0.697·24-s + 0.591·25-s − 0.702·26-s + 0.192·27-s − 1.19·28-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $1$
Analytic conductor: \(11.0664\)
Root analytic conductor: \(3.32663\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{69} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 69,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(5.390357468\)
\(L(\frac12)\) \(\approx\) \(5.390357468\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 9T \)
23 \( 1 + 529T \)
good2 \( 1 - 9.34T + 32T^{2} \)
5 \( 1 - 70.5T + 3.12e3T^{2} \)
7 \( 1 + 89.7T + 1.68e4T^{2} \)
11 \( 1 + 436.T + 1.61e5T^{2} \)
13 \( 1 + 258.T + 3.71e5T^{2} \)
17 \( 1 - 591.T + 1.41e6T^{2} \)
19 \( 1 - 1.66e3T + 2.47e6T^{2} \)
29 \( 1 + 5.12e3T + 2.05e7T^{2} \)
31 \( 1 - 1.87e3T + 2.86e7T^{2} \)
37 \( 1 - 884.T + 6.93e7T^{2} \)
41 \( 1 + 1.84e4T + 1.15e8T^{2} \)
43 \( 1 - 1.23e4T + 1.47e8T^{2} \)
47 \( 1 - 2.38e4T + 2.29e8T^{2} \)
53 \( 1 + 1.45e4T + 4.18e8T^{2} \)
59 \( 1 - 4.77e4T + 7.14e8T^{2} \)
61 \( 1 + 1.53e4T + 8.44e8T^{2} \)
67 \( 1 + 4.11e4T + 1.35e9T^{2} \)
71 \( 1 + 4.56e4T + 1.80e9T^{2} \)
73 \( 1 + 7.89e3T + 2.07e9T^{2} \)
79 \( 1 - 7.14e4T + 3.07e9T^{2} \)
83 \( 1 - 1.12e5T + 3.93e9T^{2} \)
89 \( 1 - 1.37e5T + 5.58e9T^{2} \)
97 \( 1 - 4.16e4T + 8.58e9T^{2} \)
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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

−13.57158285760811642514662672805, −13.14887498663154159641709506206, −12.06807647091299501124716761597, −10.39789656572581414984241935170, −9.398818162127121915183334138429, −7.41716477950073777319290556333, −6.02342953779757556066240162203, −5.11230797549031381038360408317, −3.33346025282642208665222941230, −2.24389398766611432919762071086, 2.24389398766611432919762071086, 3.33346025282642208665222941230, 5.11230797549031381038360408317, 6.02342953779757556066240162203, 7.41716477950073777319290556333, 9.398818162127121915183334138429, 10.39789656572581414984241935170, 12.06807647091299501124716761597, 13.14887498663154159641709506206, 13.57158285760811642514662672805

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