Properties

Label 2-69-1.1-c7-0-25
Degree $2$
Conductor $69$
Sign $-1$
Analytic cond. $21.5545$
Root an. cond. $4.64268$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 16.1·2-s + 27·3-s + 131.·4-s − 519.·5-s + 435.·6-s − 439.·7-s + 64.2·8-s + 729·9-s − 8.37e3·10-s − 1.23e3·11-s + 3.56e3·12-s − 886.·13-s − 7.08e3·14-s − 1.40e4·15-s − 1.58e4·16-s − 3.32e4·17-s + 1.17e4·18-s + 2.42e4·19-s − 6.85e4·20-s − 1.18e4·21-s − 1.99e4·22-s − 1.21e4·23-s + 1.73e3·24-s + 1.91e5·25-s − 1.42e4·26-s + 1.96e4·27-s − 5.79e4·28-s + ⋯
L(s)  = 1  + 1.42·2-s + 0.577·3-s + 1.03·4-s − 1.85·5-s + 0.822·6-s − 0.484·7-s + 0.0443·8-s + 0.333·9-s − 2.64·10-s − 0.280·11-s + 0.595·12-s − 0.111·13-s − 0.689·14-s − 1.07·15-s − 0.967·16-s − 1.64·17-s + 0.475·18-s + 0.810·19-s − 1.91·20-s − 0.279·21-s − 0.400·22-s − 0.208·23-s + 0.0256·24-s + 2.45·25-s − 0.159·26-s + 0.192·27-s − 0.499·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $-1$
Analytic conductor: \(21.5545\)
Root analytic conductor: \(4.64268\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 69,\ (\ :7/2),\ -1)\)

Particular Values

\(L(4)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 27T \)
23 \( 1 + 1.21e4T \)
good2 \( 1 - 16.1T + 128T^{2} \)
5 \( 1 + 519.T + 7.81e4T^{2} \)
7 \( 1 + 439.T + 8.23e5T^{2} \)
11 \( 1 + 1.23e3T + 1.94e7T^{2} \)
13 \( 1 + 886.T + 6.27e7T^{2} \)
17 \( 1 + 3.32e4T + 4.10e8T^{2} \)
19 \( 1 - 2.42e4T + 8.93e8T^{2} \)
29 \( 1 + 2.63e4T + 1.72e10T^{2} \)
31 \( 1 - 5.83e4T + 2.75e10T^{2} \)
37 \( 1 - 7.26e4T + 9.49e10T^{2} \)
41 \( 1 - 1.75e5T + 1.94e11T^{2} \)
43 \( 1 + 3.02e5T + 2.71e11T^{2} \)
47 \( 1 - 7.15e5T + 5.06e11T^{2} \)
53 \( 1 + 1.15e6T + 1.17e12T^{2} \)
59 \( 1 + 1.13e6T + 2.48e12T^{2} \)
61 \( 1 - 6.99e5T + 3.14e12T^{2} \)
67 \( 1 + 4.85e6T + 6.06e12T^{2} \)
71 \( 1 - 3.69e5T + 9.09e12T^{2} \)
73 \( 1 - 1.06e6T + 1.10e13T^{2} \)
79 \( 1 - 6.34e6T + 1.92e13T^{2} \)
83 \( 1 - 9.31e6T + 2.71e13T^{2} \)
89 \( 1 + 1.25e7T + 4.42e13T^{2} \)
97 \( 1 + 1.21e7T + 8.07e13T^{2} \)
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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

−12.81954688544024344554895970520, −11.94532843950750774336864336087, −11.01122518147435305558549289145, −9.061813096883941239502051841991, −7.77444249286341595585804715998, −6.63814897092251041217507209205, −4.75029310677354342908425425876, −3.84709182614320379084560114337, −2.84054475074081385744146118968, 0, 2.84054475074081385744146118968, 3.84709182614320379084560114337, 4.75029310677354342908425425876, 6.63814897092251041217507209205, 7.77444249286341595585804715998, 9.061813096883941239502051841991, 11.01122518147435305558549289145, 11.94532843950750774336864336087, 12.81954688544024344554895970520

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