Properties

Label 2-69-1.1-c5-0-11
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  + 10.7·2-s − 9·3-s + 83.9·4-s + 52.3·5-s − 96.9·6-s − 118.·7-s + 560.·8-s + 81·9-s + 564.·10-s + 741.·11-s − 755.·12-s − 542.·13-s − 1.27e3·14-s − 471.·15-s + 3.34e3·16-s − 834.·17-s + 872.·18-s − 1.30e3·19-s + 4.40e3·20-s + 1.06e3·21-s + 7.98e3·22-s + 529·23-s − 5.04e3·24-s − 380.·25-s − 5.84e3·26-s − 729·27-s − 9.93e3·28-s + ⋯
L(s)  = 1  + 1.90·2-s − 0.577·3-s + 2.62·4-s + 0.937·5-s − 1.09·6-s − 0.912·7-s + 3.09·8-s + 0.333·9-s + 1.78·10-s + 1.84·11-s − 1.51·12-s − 0.890·13-s − 1.73·14-s − 0.541·15-s + 3.26·16-s − 0.700·17-s + 0.634·18-s − 0.832·19-s + 2.45·20-s + 0.526·21-s + 3.51·22-s + 0.208·23-s − 1.78·24-s − 0.121·25-s − 1.69·26-s − 0.192·27-s − 2.39·28-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

\(L(3)\) \(\approx\) \(4.949325501\)
\(L(\frac12)\) \(\approx\) \(4.949325501\)
\(L(\frac{7}{2})\) 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 \( 1 - 10.7T + 32T^{2} \)
5 \( 1 - 52.3T + 3.12e3T^{2} \)
7 \( 1 + 118.T + 1.68e4T^{2} \)
11 \( 1 - 741.T + 1.61e5T^{2} \)
13 \( 1 + 542.T + 3.71e5T^{2} \)
17 \( 1 + 834.T + 1.41e6T^{2} \)
19 \( 1 + 1.30e3T + 2.47e6T^{2} \)
29 \( 1 - 5.53e3T + 2.05e7T^{2} \)
31 \( 1 + 7.26e3T + 2.86e7T^{2} \)
37 \( 1 + 1.04e4T + 6.93e7T^{2} \)
41 \( 1 + 4.63e3T + 1.15e8T^{2} \)
43 \( 1 + 9.20e3T + 1.47e8T^{2} \)
47 \( 1 - 1.52e4T + 2.29e8T^{2} \)
53 \( 1 + 3.62e4T + 4.18e8T^{2} \)
59 \( 1 - 6.43e3T + 7.14e8T^{2} \)
61 \( 1 + 2.63e3T + 8.44e8T^{2} \)
67 \( 1 - 2.10e4T + 1.35e9T^{2} \)
71 \( 1 - 7.46e4T + 1.80e9T^{2} \)
73 \( 1 - 8.07e4T + 2.07e9T^{2} \)
79 \( 1 - 3.91e4T + 3.07e9T^{2} \)
83 \( 1 - 1.01e5T + 3.93e9T^{2} \)
89 \( 1 - 2.24e4T + 5.58e9T^{2} \)
97 \( 1 + 2.97e4T + 8.58e9T^{2} \)
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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

−13.73060252542277886036244053644, −12.68165734366132744672456196108, −12.02896651631337138325935318147, −10.80754337700588725386412592380, −9.510942592783838027409499138020, −6.71296593317586937400900470431, −6.39984799630137542812634852709, −5.03856357092265589808621599610, −3.70839814603054667537061206146, −1.98223703096923959271753211523, 1.98223703096923959271753211523, 3.70839814603054667537061206146, 5.03856357092265589808621599610, 6.39984799630137542812634852709, 6.71296593317586937400900470431, 9.510942592783838027409499138020, 10.80754337700588725386412592380, 12.02896651631337138325935318147, 12.68165734366132744672456196108, 13.73060252542277886036244053644

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