Properties

Label 2-69-1.1-c5-0-16
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.49·2-s + 9·3-s − 19.7·4-s − 59.5·5-s + 31.4·6-s + 96.8·7-s − 180.·8-s + 81·9-s − 208.·10-s − 731.·11-s − 178.·12-s − 631.·13-s + 338.·14-s − 536.·15-s + 0.920·16-s + 75.7·17-s + 283.·18-s + 2.35·19-s + 1.17e3·20-s + 871.·21-s − 2.55e3·22-s + 529·23-s − 1.62e3·24-s + 426.·25-s − 2.20e3·26-s + 729·27-s − 1.91e3·28-s + ⋯
L(s)  = 1  + 0.617·2-s + 0.577·3-s − 0.618·4-s − 1.06·5-s + 0.356·6-s + 0.746·7-s − 0.999·8-s + 0.333·9-s − 0.658·10-s − 1.82·11-s − 0.357·12-s − 1.03·13-s + 0.461·14-s − 0.615·15-s + 0.000899·16-s + 0.0636·17-s + 0.205·18-s + 0.00149·19-s + 0.659·20-s + 0.431·21-s − 1.12·22-s + 0.208·23-s − 0.577·24-s + 0.136·25-s − 0.640·26-s + 0.192·27-s − 0.461·28-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: \(1\)
Selberg data: \((2,\ 69,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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 \( 1 - 3.49T + 32T^{2} \)
5 \( 1 + 59.5T + 3.12e3T^{2} \)
7 \( 1 - 96.8T + 1.68e4T^{2} \)
11 \( 1 + 731.T + 1.61e5T^{2} \)
13 \( 1 + 631.T + 3.71e5T^{2} \)
17 \( 1 - 75.7T + 1.41e6T^{2} \)
19 \( 1 - 2.35T + 2.47e6T^{2} \)
29 \( 1 + 2.42e3T + 2.05e7T^{2} \)
31 \( 1 + 349.T + 2.86e7T^{2} \)
37 \( 1 - 8.31e3T + 6.93e7T^{2} \)
41 \( 1 - 4.19e3T + 1.15e8T^{2} \)
43 \( 1 - 9.37e3T + 1.47e8T^{2} \)
47 \( 1 + 2.43e3T + 2.29e8T^{2} \)
53 \( 1 + 3.49e4T + 4.18e8T^{2} \)
59 \( 1 - 1.50e4T + 7.14e8T^{2} \)
61 \( 1 - 987.T + 8.44e8T^{2} \)
67 \( 1 - 4.43e4T + 1.35e9T^{2} \)
71 \( 1 + 3.39e4T + 1.80e9T^{2} \)
73 \( 1 + 4.24e4T + 2.07e9T^{2} \)
79 \( 1 - 5.11e4T + 3.07e9T^{2} \)
83 \( 1 + 8.84e4T + 3.93e9T^{2} \)
89 \( 1 + 3.40e4T + 5.58e9T^{2} \)
97 \( 1 + 1.56e5T + 8.58e9T^{2} \)
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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

−13.18714788501579423187305885888, −12.39929156605579735569153989201, −11.12279833255946615410651079087, −9.683486347078546532490312006770, −8.219743289068679694242388266965, −7.60390341143152664273956425077, −5.27476984152840704018611085494, −4.29743650348217054499033465717, −2.79437311095028638023727714666, 0, 2.79437311095028638023727714666, 4.29743650348217054499033465717, 5.27476984152840704018611085494, 7.60390341143152664273956425077, 8.219743289068679694242388266965, 9.683486347078546532490312006770, 11.12279833255946615410651079087, 12.39929156605579735569153989201, 13.18714788501579423187305885888

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