Properties

Label 2-3e5-1.1-c1-0-7
Degree $2$
Conductor $243$
Sign $-1$
Analytic cond. $1.94036$
Root an. cond. $1.39296$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.34·2-s − 0.184·4-s − 1.65·5-s + 2.41·7-s + 2.94·8-s + 2.22·10-s − 5.94·11-s − 3.22·13-s − 3.24·14-s − 3.59·16-s − 3·17-s − 6.63·19-s + 0.305·20-s + 8.00·22-s + 2.94·23-s − 2.26·25-s + 4.34·26-s − 0.445·28-s + 1.29·29-s − 0.588·31-s − 1.04·32-s + 4.04·34-s − 3.98·35-s + 0.0418·37-s + 8.94·38-s − 4.86·40-s + 4.90·41-s + ⋯
L(s)  = 1  − 0.952·2-s − 0.0923·4-s − 0.739·5-s + 0.911·7-s + 1.04·8-s + 0.704·10-s − 1.79·11-s − 0.894·13-s − 0.868·14-s − 0.899·16-s − 0.727·17-s − 1.52·19-s + 0.0682·20-s + 1.70·22-s + 0.613·23-s − 0.453·25-s + 0.852·26-s − 0.0842·28-s + 0.239·29-s − 0.105·31-s − 0.184·32-s + 0.693·34-s − 0.673·35-s + 0.00688·37-s + 1.45·38-s − 0.769·40-s + 0.765·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(243\)    =    \(3^{5}\)
Sign: $-1$
Analytic conductor: \(1.94036\)
Root analytic conductor: \(1.39296\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 243,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.34T + 2T^{2} \)
5 \( 1 + 1.65T + 5T^{2} \)
7 \( 1 - 2.41T + 7T^{2} \)
11 \( 1 + 5.94T + 11T^{2} \)
13 \( 1 + 3.22T + 13T^{2} \)
17 \( 1 + 3T + 17T^{2} \)
19 \( 1 + 6.63T + 19T^{2} \)
23 \( 1 - 2.94T + 23T^{2} \)
29 \( 1 - 1.29T + 29T^{2} \)
31 \( 1 + 0.588T + 31T^{2} \)
37 \( 1 - 0.0418T + 37T^{2} \)
41 \( 1 - 4.90T + 41T^{2} \)
43 \( 1 + 5.18T + 43T^{2} \)
47 \( 1 - 3.73T + 47T^{2} \)
53 \( 1 + 11.6T + 53T^{2} \)
59 \( 1 - 7.34T + 59T^{2} \)
61 \( 1 - 11.0T + 61T^{2} \)
67 \( 1 - 1.85T + 67T^{2} \)
71 \( 1 - 5.51T + 71T^{2} \)
73 \( 1 - 5.55T + 73T^{2} \)
79 \( 1 + 3.78T + 79T^{2} \)
83 \( 1 + 3.98T + 83T^{2} \)
89 \( 1 + 8.15T + 89T^{2} \)
97 \( 1 - 0.260T + 97T^{2} \)
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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

−11.21094927693477080970868832223, −10.69805872945867638656025958998, −9.698218293970934472866009874913, −8.387826227475013430850662966704, −8.041914134602312911887435922690, −7.07973079361832528250324311605, −5.15289469433362793276361238788, −4.35238690733375582841137514901, −2.27520788694591860789670521352, 0, 2.27520788694591860789670521352, 4.35238690733375582841137514901, 5.15289469433362793276361238788, 7.07973079361832528250324311605, 8.041914134602312911887435922690, 8.387826227475013430850662966704, 9.698218293970934472866009874913, 10.69805872945867638656025958998, 11.21094927693477080970868832223

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