Properties

Label 2-69-1.1-c5-0-2
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  − 3.54·2-s + 9·3-s − 19.4·4-s − 92.1·5-s − 31.8·6-s − 8.98·7-s + 182.·8-s + 81·9-s + 326.·10-s + 612.·11-s − 175.·12-s − 496.·13-s + 31.8·14-s − 829.·15-s − 23.4·16-s + 745.·17-s − 286.·18-s + 1.29e3·19-s + 1.79e3·20-s − 80.9·21-s − 2.17e3·22-s − 529·23-s + 1.64e3·24-s + 5.36e3·25-s + 1.75e3·26-s + 729·27-s + 174.·28-s + ⋯
L(s)  = 1  − 0.626·2-s + 0.577·3-s − 0.607·4-s − 1.64·5-s − 0.361·6-s − 0.0693·7-s + 1.00·8-s + 0.333·9-s + 1.03·10-s + 1.52·11-s − 0.350·12-s − 0.814·13-s + 0.0434·14-s − 0.951·15-s − 0.0228·16-s + 0.625·17-s − 0.208·18-s + 0.825·19-s + 1.00·20-s − 0.0400·21-s − 0.956·22-s − 0.208·23-s + 0.581·24-s + 1.71·25-s + 0.510·26-s + 0.192·27-s + 0.0421·28-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

\(L(3)\) \(\approx\) \(0.9664536085\)
\(L(\frac12)\) \(\approx\) \(0.9664536085\)
\(L(\frac{7}{2})\) 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 \( 1 + 3.54T + 32T^{2} \)
5 \( 1 + 92.1T + 3.12e3T^{2} \)
7 \( 1 + 8.98T + 1.68e4T^{2} \)
11 \( 1 - 612.T + 1.61e5T^{2} \)
13 \( 1 + 496.T + 3.71e5T^{2} \)
17 \( 1 - 745.T + 1.41e6T^{2} \)
19 \( 1 - 1.29e3T + 2.47e6T^{2} \)
29 \( 1 - 7.75e3T + 2.05e7T^{2} \)
31 \( 1 + 1.06e3T + 2.86e7T^{2} \)
37 \( 1 + 840.T + 6.93e7T^{2} \)
41 \( 1 - 9.82e3T + 1.15e8T^{2} \)
43 \( 1 + 1.35e4T + 1.47e8T^{2} \)
47 \( 1 - 2.04e4T + 2.29e8T^{2} \)
53 \( 1 - 4.81e3T + 4.18e8T^{2} \)
59 \( 1 - 3.05e4T + 7.14e8T^{2} \)
61 \( 1 - 9.53T + 8.44e8T^{2} \)
67 \( 1 - 9.99e3T + 1.35e9T^{2} \)
71 \( 1 + 7.37e4T + 1.80e9T^{2} \)
73 \( 1 - 5.51e4T + 2.07e9T^{2} \)
79 \( 1 - 6.23e4T + 3.07e9T^{2} \)
83 \( 1 + 8.24e4T + 3.93e9T^{2} \)
89 \( 1 - 9.51e4T + 5.58e9T^{2} \)
97 \( 1 - 4.71e4T + 8.58e9T^{2} \)
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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

−14.02444355149951149352689985262, −12.42782918073078516525213616656, −11.63122173781117147315355718384, −10.05818148688829285093997454709, −8.985329741924781868167586931587, −8.037280753225211284468601151860, −7.12167511553267171037756707280, −4.57896778918635400770656627339, −3.53511996991014255088625359725, −0.860939740144385835086401300255, 0.860939740144385835086401300255, 3.53511996991014255088625359725, 4.57896778918635400770656627339, 7.12167511553267171037756707280, 8.037280753225211284468601151860, 8.985329741924781868167586931587, 10.05818148688829285093997454709, 11.63122173781117147315355718384, 12.42782918073078516525213616656, 14.02444355149951149352689985262

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