Properties

Label 2-3e5-9.7-c1-0-1
Degree $2$
Conductor $243$
Sign $-0.939 + 0.342i$
Analytic cond. $1.94036$
Root an. cond. $1.39296$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.22 + 2.12i)2-s + (−1.99 − 3.46i)4-s + (1.22 + 2.12i)5-s + (−1 + 1.73i)7-s + 4.89·8-s − 5.99·10-s + (−1.22 + 2.12i)11-s + (0.5 + 0.866i)13-s + (−2.44 − 4.24i)14-s + (−1.99 + 3.46i)16-s − 7.34·17-s − 19-s + (4.89 − 8.48i)20-s + (−2.99 − 5.19i)22-s + (1.22 + 2.12i)23-s + ⋯
L(s)  = 1  + (−0.866 + 1.49i)2-s + (−0.999 − 1.73i)4-s + (0.547 + 0.948i)5-s + (−0.377 + 0.654i)7-s + 1.73·8-s − 1.89·10-s + (−0.369 + 0.639i)11-s + (0.138 + 0.240i)13-s + (−0.654 − 1.13i)14-s + (−0.499 + 0.866i)16-s − 1.78·17-s − 0.229·19-s + (1.09 − 1.89i)20-s + (−0.639 − 1.10i)22-s + (0.255 + 0.442i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(243\)    =    \(3^{5}\)
Sign: $-0.939 + 0.342i$
Analytic conductor: \(1.94036\)
Root analytic conductor: \(1.39296\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{243} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 243,\ (\ :1/2),\ -0.939 + 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.107972 - 0.612345i\)
\(L(\frac12)\) \(\approx\) \(0.107972 - 0.612345i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.22 - 2.12i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-1.22 - 2.12i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (1 - 1.73i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.22 - 2.12i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 7.34T + 17T^{2} \)
19 \( 1 + T + 19T^{2} \)
23 \( 1 + (-1.22 - 2.12i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.44 + 4.24i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 8T + 37T^{2} \)
41 \( 1 + (2.44 + 4.24i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.5 - 9.52i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.89 - 8.48i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 7.34T + 53T^{2} \)
59 \( 1 + (-1.22 - 2.12i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.5 - 4.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.5 - 6.06i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 7.34T + 71T^{2} \)
73 \( 1 - 11T + 73T^{2} \)
79 \( 1 + (-3.5 + 6.06i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6.12 + 10.6i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + (-3.5 + 6.06i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.94579562529682362176775151035, −11.39698416724721359424354471746, −10.29693989926362402050292180491, −9.513201901280145981191689444283, −8.713901262568728525201304567929, −7.58170957285297468948198693112, −6.55616342148556771700564567083, −6.14358600324312855867809649611, −4.73850103356726209616932144102, −2.46194146608919366029175966085, 0.65273118982897499701926485849, 2.21408349514242148579804993749, 3.66230767495311126607476919101, 4.95620541544807605412390087187, 6.64378270990586805387273158794, 8.288520553585180625172940386799, 8.853295464205832290362658378422, 9.762936483147107092606900401252, 10.63160886824101681585272475530, 11.30405443059885596771913874614

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