Properties

Label 2-3e2-1.1-c37-0-7
Degree $2$
Conductor $9$
Sign $1$
Analytic cond. $78.0426$
Root an. cond. $8.83417$
Motivic weight $37$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.86e5·2-s − 5.56e10·4-s + 8.29e12·5-s + 1.97e15·7-s + 5.52e16·8-s − 2.37e18·10-s + 2.57e19·11-s + 5.42e20·13-s − 5.64e20·14-s − 8.14e21·16-s + 3.52e22·17-s + 6.82e23·19-s − 4.61e23·20-s − 7.35e24·22-s + 8.19e22·23-s − 3.91e24·25-s − 1.55e26·26-s − 1.09e26·28-s + 1.51e27·29-s + 2.60e27·31-s − 5.25e27·32-s − 1.00e28·34-s + 1.63e28·35-s − 1.30e29·37-s − 1.95e29·38-s + 4.58e29·40-s + 4.07e29·41-s + ⋯
L(s)  = 1  − 0.771·2-s − 0.404·4-s + 0.972·5-s + 0.458·7-s + 1.08·8-s − 0.750·10-s + 1.39·11-s + 1.33·13-s − 0.353·14-s − 0.431·16-s + 0.608·17-s + 1.50·19-s − 0.393·20-s − 1.07·22-s + 0.00526·23-s − 0.0538·25-s − 1.03·26-s − 0.185·28-s + 1.34·29-s + 0.670·31-s − 0.751·32-s − 0.469·34-s + 0.445·35-s − 1.26·37-s − 1.15·38-s + 1.05·40-s + 0.593·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(38-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+37/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9\)    =    \(3^{2}\)
Sign: $1$
Analytic conductor: \(78.0426\)
Root analytic conductor: \(8.83417\)
Motivic weight: \(37\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 9,\ (\ :37/2),\ 1)\)

Particular Values

\(L(19)\) \(\approx\) \(2.202291072\)
\(L(\frac12)\) \(\approx\) \(2.202291072\)
\(L(\frac{39}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
good2 \( 1 + 2.86e5T + 1.37e11T^{2} \)
5 \( 1 - 8.29e12T + 7.27e25T^{2} \)
7 \( 1 - 1.97e15T + 1.85e31T^{2} \)
11 \( 1 - 2.57e19T + 3.40e38T^{2} \)
13 \( 1 - 5.42e20T + 1.64e41T^{2} \)
17 \( 1 - 3.52e22T + 3.36e45T^{2} \)
19 \( 1 - 6.82e23T + 2.06e47T^{2} \)
23 \( 1 - 8.19e22T + 2.42e50T^{2} \)
29 \( 1 - 1.51e27T + 1.28e54T^{2} \)
31 \( 1 - 2.60e27T + 1.51e55T^{2} \)
37 \( 1 + 1.30e29T + 1.05e58T^{2} \)
41 \( 1 - 4.07e29T + 4.70e59T^{2} \)
43 \( 1 + 2.92e30T + 2.74e60T^{2} \)
47 \( 1 + 3.58e30T + 7.37e61T^{2} \)
53 \( 1 + 3.56e31T + 6.28e63T^{2} \)
59 \( 1 - 3.03e32T + 3.32e65T^{2} \)
61 \( 1 - 1.16e33T + 1.14e66T^{2} \)
67 \( 1 + 2.44e33T + 3.67e67T^{2} \)
71 \( 1 + 6.30e33T + 3.13e68T^{2} \)
73 \( 1 - 1.02e34T + 8.76e68T^{2} \)
79 \( 1 - 1.20e35T + 1.63e70T^{2} \)
83 \( 1 - 3.26e35T + 1.01e71T^{2} \)
89 \( 1 - 1.56e36T + 1.34e72T^{2} \)
97 \( 1 + 1.07e36T + 3.24e73T^{2} \)
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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

−13.56982560306490774743633173202, −11.69793327349586923753835272384, −10.18487672862619229966687375147, −9.240843808415960017862893798652, −8.181767095037083665039383790276, −6.50631012254476021640822163523, −5.10942836408317068989441894151, −3.58817879819539040188777686579, −1.55118135216588969185328888964, −1.01378084956321360107102645348, 1.01378084956321360107102645348, 1.55118135216588969185328888964, 3.58817879819539040188777686579, 5.10942836408317068989441894151, 6.50631012254476021640822163523, 8.181767095037083665039383790276, 9.240843808415960017862893798652, 10.18487672862619229966687375147, 11.69793327349586923753835272384, 13.56982560306490774743633173202

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