Properties

Label 2-69-1.1-c5-0-5
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  − 9.32·2-s + 9·3-s + 54.9·4-s + 99.7·5-s − 83.9·6-s + 125.·7-s − 214.·8-s + 81·9-s − 930.·10-s + 177.·11-s + 494.·12-s − 919.·13-s − 1.17e3·14-s + 897.·15-s + 240.·16-s + 1.56e3·17-s − 755.·18-s + 447.·19-s + 5.48e3·20-s + 1.13e3·21-s − 1.65e3·22-s − 529·23-s − 1.93e3·24-s + 6.81e3·25-s + 8.57e3·26-s + 729·27-s + 6.92e3·28-s + ⋯
L(s)  = 1  − 1.64·2-s + 0.577·3-s + 1.71·4-s + 1.78·5-s − 0.951·6-s + 0.971·7-s − 1.18·8-s + 0.333·9-s − 2.94·10-s + 0.442·11-s + 0.992·12-s − 1.50·13-s − 1.60·14-s + 1.02·15-s + 0.235·16-s + 1.31·17-s − 0.549·18-s + 0.284·19-s + 3.06·20-s + 0.561·21-s − 0.730·22-s − 0.208·23-s − 0.684·24-s + 2.18·25-s + 2.48·26-s + 0.192·27-s + 1.67·28-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

\(L(3)\) \(\approx\) \(1.460469041\)
\(L(\frac12)\) \(\approx\) \(1.460469041\)
\(L(\frac{7}{2})\) 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 \( 1 + 9.32T + 32T^{2} \)
5 \( 1 - 99.7T + 3.12e3T^{2} \)
7 \( 1 - 125.T + 1.68e4T^{2} \)
11 \( 1 - 177.T + 1.61e5T^{2} \)
13 \( 1 + 919.T + 3.71e5T^{2} \)
17 \( 1 - 1.56e3T + 1.41e6T^{2} \)
19 \( 1 - 447.T + 2.47e6T^{2} \)
29 \( 1 + 1.29e3T + 2.05e7T^{2} \)
31 \( 1 + 6.65e3T + 2.86e7T^{2} \)
37 \( 1 + 8.05e3T + 6.93e7T^{2} \)
41 \( 1 - 1.79e4T + 1.15e8T^{2} \)
43 \( 1 - 1.07e4T + 1.47e8T^{2} \)
47 \( 1 + 1.78e4T + 2.29e8T^{2} \)
53 \( 1 + 2.03e4T + 4.18e8T^{2} \)
59 \( 1 - 2.84e4T + 7.14e8T^{2} \)
61 \( 1 - 1.62e4T + 8.44e8T^{2} \)
67 \( 1 - 5.41e4T + 1.35e9T^{2} \)
71 \( 1 - 4.25e4T + 1.80e9T^{2} \)
73 \( 1 - 4.09e4T + 2.07e9T^{2} \)
79 \( 1 + 2.47e4T + 3.07e9T^{2} \)
83 \( 1 + 3.08e4T + 3.93e9T^{2} \)
89 \( 1 + 6.65e4T + 5.58e9T^{2} \)
97 \( 1 + 1.04e5T + 8.58e9T^{2} \)
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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

−14.16526574554128143465529243572, −12.53128808663571473662143967899, −10.97118806892819696068065747106, −9.746008404494026706243769606036, −9.486359367968845420967693155008, −8.131878632539373894394789052338, −7.04942233489018741394881458200, −5.35898668893587231901003368819, −2.33644919905903927464820475449, −1.37838222334649328233933981507, 1.37838222334649328233933981507, 2.33644919905903927464820475449, 5.35898668893587231901003368819, 7.04942233489018741394881458200, 8.131878632539373894394789052338, 9.486359367968845420967693155008, 9.746008404494026706243769606036, 10.97118806892819696068065747106, 12.53128808663571473662143967899, 14.16526574554128143465529243572

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