Properties

Label 2-3e2-1.1-c23-0-6
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  + 4.78e3·2-s + 1.44e7·4-s + 1.42e8·5-s + 6.49e9·7-s + 2.92e10·8-s + 6.81e11·10-s − 6.56e11·11-s − 1.03e13·13-s + 3.10e13·14-s + 1.81e13·16-s + 2.36e14·17-s + 5.53e14·19-s + 2.06e15·20-s − 3.14e15·22-s + 3.36e15·23-s + 8.39e15·25-s − 4.93e16·26-s + 9.41e16·28-s − 8.75e15·29-s − 1.61e17·31-s − 1.58e17·32-s + 1.12e18·34-s + 9.25e17·35-s + 1.42e18·37-s + 2.64e18·38-s + 4.16e18·40-s − 4.86e18·41-s + ⋯
L(s)  = 1  + 1.65·2-s + 1.72·4-s + 1.30·5-s + 1.24·7-s + 1.20·8-s + 2.15·10-s − 0.694·11-s − 1.59·13-s + 2.05·14-s + 0.258·16-s + 1.67·17-s + 1.08·19-s + 2.25·20-s − 1.14·22-s + 0.736·23-s + 0.703·25-s − 2.63·26-s + 2.14·28-s − 0.133·29-s − 1.14·31-s − 0.775·32-s + 2.76·34-s + 1.62·35-s + 1.31·37-s + 1.79·38-s + 1.57·40-s − 1.38·41-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

\(L(12)\) \(\approx\) \(6.857959030\)
\(L(\frac12)\) \(\approx\) \(6.857959030\)
\(L(\frac{25}{2})\) 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 \( 1 - 4.78e3T + 8.38e6T^{2} \)
5 \( 1 - 1.42e8T + 1.19e16T^{2} \)
7 \( 1 - 6.49e9T + 2.73e19T^{2} \)
11 \( 1 + 6.56e11T + 8.95e23T^{2} \)
13 \( 1 + 1.03e13T + 4.17e25T^{2} \)
17 \( 1 - 2.36e14T + 1.99e28T^{2} \)
19 \( 1 - 5.53e14T + 2.57e29T^{2} \)
23 \( 1 - 3.36e15T + 2.08e31T^{2} \)
29 \( 1 + 8.75e15T + 4.31e33T^{2} \)
31 \( 1 + 1.61e17T + 2.00e34T^{2} \)
37 \( 1 - 1.42e18T + 1.17e36T^{2} \)
41 \( 1 + 4.86e18T + 1.24e37T^{2} \)
43 \( 1 + 3.63e18T + 3.71e37T^{2} \)
47 \( 1 + 1.28e19T + 2.87e38T^{2} \)
53 \( 1 - 1.00e19T + 4.55e39T^{2} \)
59 \( 1 - 1.29e19T + 5.36e40T^{2} \)
61 \( 1 - 1.60e20T + 1.15e41T^{2} \)
67 \( 1 - 3.50e20T + 9.99e41T^{2} \)
71 \( 1 + 2.05e21T + 3.79e42T^{2} \)
73 \( 1 + 9.56e20T + 7.18e42T^{2} \)
79 \( 1 + 4.09e21T + 4.42e43T^{2} \)
83 \( 1 + 1.62e22T + 1.37e44T^{2} \)
89 \( 1 + 2.59e21T + 6.85e44T^{2} \)
97 \( 1 - 2.98e22T + 4.96e45T^{2} \)
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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

−14.81767395573239611790427890550, −14.20487748902096579784513956383, −12.95915890874217166535653333171, −11.64773203896450940576564229754, −9.933370612220743052987728880528, −7.43680692351053601789210555530, −5.50426009732325169538010563041, −5.00209455176749215026207004799, −2.90822122051428103170727562707, −1.68995568615001132665999226100, 1.68995568615001132665999226100, 2.90822122051428103170727562707, 5.00209455176749215026207004799, 5.50426009732325169538010563041, 7.43680692351053601789210555530, 9.933370612220743052987728880528, 11.64773203896450940576564229754, 12.95915890874217166535653333171, 14.20487748902096579784513956383, 14.81767395573239611790427890550

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