Properties

Label 2-69-1.1-c7-0-6
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 16.1·2-s + 27·3-s + 132.·4-s + 118.·5-s − 436.·6-s + 738.·7-s − 80.6·8-s + 729·9-s − 1.91e3·10-s + 4.27e3·11-s + 3.59e3·12-s − 8.30·13-s − 1.19e4·14-s + 3.19e3·15-s − 1.57e4·16-s − 9.64e3·17-s − 1.17e4·18-s − 1.44e4·19-s + 1.57e4·20-s + 1.99e4·21-s − 6.91e4·22-s + 1.21e4·23-s − 2.17e3·24-s − 6.41e4·25-s + 134.·26-s + 1.96e4·27-s + 9.82e4·28-s + ⋯
L(s)  = 1  − 1.42·2-s + 0.577·3-s + 1.03·4-s + 0.423·5-s − 0.824·6-s + 0.813·7-s − 0.0557·8-s + 0.333·9-s − 0.604·10-s + 0.969·11-s + 0.599·12-s − 0.00104·13-s − 1.16·14-s + 0.244·15-s − 0.959·16-s − 0.475·17-s − 0.475·18-s − 0.483·19-s + 0.439·20-s + 0.469·21-s − 1.38·22-s + 0.208·23-s − 0.0321·24-s − 0.820·25-s + 0.00149·26-s + 0.192·27-s + 0.845·28-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: \(0\)
Selberg data: \((2,\ 69,\ (\ :7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(1.413369640\)
\(L(\frac12)\) \(\approx\) \(1.413369640\)
\(L(\frac{9}{2})\) 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 \( 1 - 118.T + 7.81e4T^{2} \)
7 \( 1 - 738.T + 8.23e5T^{2} \)
11 \( 1 - 4.27e3T + 1.94e7T^{2} \)
13 \( 1 + 8.30T + 6.27e7T^{2} \)
17 \( 1 + 9.64e3T + 4.10e8T^{2} \)
19 \( 1 + 1.44e4T + 8.93e8T^{2} \)
29 \( 1 - 1.98e5T + 1.72e10T^{2} \)
31 \( 1 - 8.05e4T + 2.75e10T^{2} \)
37 \( 1 - 4.16e5T + 9.49e10T^{2} \)
41 \( 1 - 2.34e5T + 1.94e11T^{2} \)
43 \( 1 + 3.62e5T + 2.71e11T^{2} \)
47 \( 1 - 8.78e5T + 5.06e11T^{2} \)
53 \( 1 - 9.71e5T + 1.17e12T^{2} \)
59 \( 1 + 9.16e5T + 2.48e12T^{2} \)
61 \( 1 + 4.44e4T + 3.14e12T^{2} \)
67 \( 1 + 1.44e6T + 6.06e12T^{2} \)
71 \( 1 - 2.27e6T + 9.09e12T^{2} \)
73 \( 1 - 2.56e6T + 1.10e13T^{2} \)
79 \( 1 + 1.09e6T + 1.92e13T^{2} \)
83 \( 1 - 3.98e6T + 2.71e13T^{2} \)
89 \( 1 - 7.23e6T + 4.42e13T^{2} \)
97 \( 1 + 8.46e5T + 8.07e13T^{2} \)
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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

−13.48291062543730029803601725326, −11.82015057703015761619427725334, −10.68675308360815145206860088849, −9.605975531846220808362365433485, −8.733363503651834486595350141329, −7.83904094196315606154975668989, −6.51631880248209192975429408235, −4.41803420991700869600457114297, −2.22168486101543150311364625944, −1.04312081132085226704607552122, 1.04312081132085226704607552122, 2.22168486101543150311364625944, 4.41803420991700869600457114297, 6.51631880248209192975429408235, 7.83904094196315606154975668989, 8.733363503651834486595350141329, 9.605975531846220808362365433485, 10.68675308360815145206860088849, 11.82015057703015761619427725334, 13.48291062543730029803601725326

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