Properties

Label 2-69-1.1-c7-0-14
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  − 3.24·2-s − 27·3-s − 117.·4-s + 168.·5-s + 87.6·6-s + 149.·7-s + 796.·8-s + 729·9-s − 545.·10-s − 18.5·11-s + 3.17e3·12-s + 1.02e4·13-s − 486.·14-s − 4.53e3·15-s + 1.24e4·16-s − 2.59e4·17-s − 2.36e3·18-s − 1.61e4·19-s − 1.97e4·20-s − 4.04e3·21-s + 60.1·22-s + 1.21e4·23-s − 2.15e4·24-s − 4.98e4·25-s − 3.31e4·26-s − 1.96e4·27-s − 1.75e4·28-s + ⋯
L(s)  = 1  − 0.286·2-s − 0.577·3-s − 0.917·4-s + 0.601·5-s + 0.165·6-s + 0.165·7-s + 0.550·8-s + 0.333·9-s − 0.172·10-s − 0.00420·11-s + 0.529·12-s + 1.28·13-s − 0.0473·14-s − 0.347·15-s + 0.759·16-s − 1.27·17-s − 0.0956·18-s − 0.540·19-s − 0.551·20-s − 0.0953·21-s + 0.00120·22-s + 0.208·23-s − 0.317·24-s − 0.638·25-s − 0.369·26-s − 0.192·27-s − 0.151·28-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: $\chi_{69} (1, \cdot )$
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 \( 1 + 3.24T + 128T^{2} \)
5 \( 1 - 168.T + 7.81e4T^{2} \)
7 \( 1 - 149.T + 8.23e5T^{2} \)
11 \( 1 + 18.5T + 1.94e7T^{2} \)
13 \( 1 - 1.02e4T + 6.27e7T^{2} \)
17 \( 1 + 2.59e4T + 4.10e8T^{2} \)
19 \( 1 + 1.61e4T + 8.93e8T^{2} \)
29 \( 1 + 9.28e4T + 1.72e10T^{2} \)
31 \( 1 + 5.22e4T + 2.75e10T^{2} \)
37 \( 1 + 7.66e4T + 9.49e10T^{2} \)
41 \( 1 - 1.08e5T + 1.94e11T^{2} \)
43 \( 1 + 7.91e5T + 2.71e11T^{2} \)
47 \( 1 + 9.92e5T + 5.06e11T^{2} \)
53 \( 1 - 1.60e6T + 1.17e12T^{2} \)
59 \( 1 - 7.33e5T + 2.48e12T^{2} \)
61 \( 1 - 1.14e5T + 3.14e12T^{2} \)
67 \( 1 - 4.47e5T + 6.06e12T^{2} \)
71 \( 1 + 6.71e5T + 9.09e12T^{2} \)
73 \( 1 + 4.06e5T + 1.10e13T^{2} \)
79 \( 1 - 8.83e5T + 1.92e13T^{2} \)
83 \( 1 + 4.07e6T + 2.71e13T^{2} \)
89 \( 1 - 5.07e6T + 4.42e13T^{2} \)
97 \( 1 + 1.23e6T + 8.07e13T^{2} \)
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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

−13.07680974348424921068935612608, −11.42419753320104578402698196752, −10.42207266710544950036488433000, −9.271925824460741390156780114522, −8.276826841433842585233499034133, −6.53968313003047788688451267022, −5.31078086548366039912166637434, −3.98640432522017386256303090141, −1.59959296699392714695741523845, 0, 1.59959296699392714695741523845, 3.98640432522017386256303090141, 5.31078086548366039912166637434, 6.53968313003047788688451267022, 8.276826841433842585233499034133, 9.271925824460741390156780114522, 10.42207266710544950036488433000, 11.42419753320104578402698196752, 13.07680974348424921068935612608

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