Properties

Label 2-3e2-1.1-c23-0-2
Degree $2$
Conductor $9$
Sign $1$
Analytic cond. $30.1683$
Root an. cond. $5.49257$
Motivic weight $23$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.85e3·2-s − 4.94e6·4-s + 1.25e8·5-s − 2.21e9·7-s + 2.47e10·8-s − 2.32e11·10-s − 9.54e11·11-s + 3.54e12·13-s + 4.11e12·14-s − 4.48e12·16-s − 1.97e14·17-s − 1.41e14·19-s − 6.18e14·20-s + 1.77e15·22-s + 1.35e15·23-s + 3.73e15·25-s − 6.57e15·26-s + 1.09e16·28-s + 1.17e17·29-s + 2.61e17·31-s − 1.99e17·32-s + 3.66e17·34-s − 2.77e17·35-s + 1.63e18·37-s + 2.63e17·38-s + 3.09e18·40-s − 1.39e18·41-s + ⋯
L(s)  = 1  − 0.640·2-s − 0.589·4-s + 1.14·5-s − 0.423·7-s + 1.01·8-s − 0.734·10-s − 1.00·11-s + 0.548·13-s + 0.271·14-s − 0.0637·16-s − 1.39·17-s − 0.279·19-s − 0.675·20-s + 0.646·22-s + 0.297·23-s + 0.313·25-s − 0.351·26-s + 0.249·28-s + 1.78·29-s + 1.84·31-s − 0.977·32-s + 0.896·34-s − 0.485·35-s + 1.50·37-s + 0.178·38-s + 1.16·40-s − 0.397·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+23/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9\)    =    \(3^{2}\)
Sign: $1$
Analytic conductor: \(30.1683\)
Root analytic conductor: \(5.49257\)
Motivic weight: \(23\)
Rational: no
Arithmetic: yes
Character: $\chi_{9} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 9,\ (\ :23/2),\ 1)\)

Particular Values

\(L(12)\) \(\approx\) \(1.239074675\)
\(L(\frac12)\) \(\approx\) \(1.239074675\)
\(L(\frac{25}{2})\) 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 \( 1 + 1.85e3T + 8.38e6T^{2} \)
5 \( 1 - 1.25e8T + 1.19e16T^{2} \)
7 \( 1 + 2.21e9T + 2.73e19T^{2} \)
11 \( 1 + 9.54e11T + 8.95e23T^{2} \)
13 \( 1 - 3.54e12T + 4.17e25T^{2} \)
17 \( 1 + 1.97e14T + 1.99e28T^{2} \)
19 \( 1 + 1.41e14T + 2.57e29T^{2} \)
23 \( 1 - 1.35e15T + 2.08e31T^{2} \)
29 \( 1 - 1.17e17T + 4.31e33T^{2} \)
31 \( 1 - 2.61e17T + 2.00e34T^{2} \)
37 \( 1 - 1.63e18T + 1.17e36T^{2} \)
41 \( 1 + 1.39e18T + 1.24e37T^{2} \)
43 \( 1 - 7.83e18T + 3.71e37T^{2} \)
47 \( 1 - 1.10e19T + 2.87e38T^{2} \)
53 \( 1 + 6.84e19T + 4.55e39T^{2} \)
59 \( 1 - 3.80e20T + 5.36e40T^{2} \)
61 \( 1 + 2.54e20T + 1.15e41T^{2} \)
67 \( 1 - 7.58e20T + 9.99e41T^{2} \)
71 \( 1 + 1.65e21T + 3.79e42T^{2} \)
73 \( 1 - 3.73e21T + 7.18e42T^{2} \)
79 \( 1 - 3.30e21T + 4.42e43T^{2} \)
83 \( 1 - 1.73e22T + 1.37e44T^{2} \)
89 \( 1 - 7.98e20T + 6.85e44T^{2} \)
97 \( 1 - 9.51e20T + 4.96e45T^{2} \)
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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

−15.83682602298986040261874128270, −13.80899616419917404510574187036, −13.05635826924556953125885823201, −10.62391160345168032842104275791, −9.578566846220428344672283432447, −8.313996480442497466888002669445, −6.31662009895272285792141227640, −4.67782008855789247372353534494, −2.44485877300628009411234574490, −0.78015644226417169727662273490, 0.78015644226417169727662273490, 2.44485877300628009411234574490, 4.67782008855789247372353534494, 6.31662009895272285792141227640, 8.313996480442497466888002669445, 9.578566846220428344672283432447, 10.62391160345168032842104275791, 13.05635826924556953125885823201, 13.80899616419917404510574187036, 15.83682602298986040261874128270

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