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Properties

Label 2-3e5-9.7-c1-0-7
Degree $2$
Conductor $243$
Sign $0.766 + 0.642i$
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  + (0.866 − 1.5i)2-s + (−0.5 − 0.866i)4-s + (1.73 + 3i)5-s + (0.5 − 0.866i)7-s + 1.73·8-s + 6·10-s + (−1.73 + 3i)11-s + (−2.5 − 4.33i)13-s + (−0.866 − 1.5i)14-s + (2.49 − 4.33i)16-s − 19-s + (1.73 − 3i)20-s + (3 + 5.19i)22-s + (−3.46 − 6i)23-s + (−3.5 + 6.06i)25-s − 8.66·26-s + ⋯
L(s)  = 1  + (0.612 − 1.06i)2-s + (−0.250 − 0.433i)4-s + (0.774 + 1.34i)5-s + (0.188 − 0.327i)7-s + 0.612·8-s + 1.89·10-s + (−0.522 + 0.904i)11-s + (−0.693 − 1.20i)13-s + (−0.231 − 0.400i)14-s + (0.624 − 1.08i)16-s − 0.229·19-s + (0.387 − 0.670i)20-s + (0.639 + 1.10i)22-s + (−0.722 − 1.25i)23-s + (−0.700 + 1.21i)25-s − 1.69·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(243\)    =    \(3^{5}\)
Sign: $0.766 + 0.642i$
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: \((2,\ 243,\ (\ :1/2),\ 0.766 + 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.79165 - 0.652107i\)
\(L(\frac12)\) \(\approx\) \(1.79165 - 0.652107i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 + 1.5i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-1.73 - 3i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-0.5 + 0.866i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.73 - 3i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.5 + 4.33i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + T + 19T^{2} \)
23 \( 1 + (3.46 + 6i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.73 - 3i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.5 + 4.33i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + T + 37T^{2} \)
41 \( 1 + (-1.73 - 3i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.73 - 3i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 10.3T + 53T^{2} \)
59 \( 1 + (-1.73 - 3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1 - 1.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4 + 6.92i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 10.3T + 71T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.46 + 6i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 10.3T + 89T^{2} \)
97 \( 1 + (8.5 - 14.7i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.08927992941335578009892212926, −10.78640462950096596729410111834, −10.48038938322974848562198398821, −9.735941098852421848983030941166, −7.82704421819917689819318639729, −7.02881533533033942315144179759, −5.64648580688680913343882915749, −4.37094827743046545745590114849, −2.94595583608623063452189652700, −2.17825215517418139029996194373, 1.83041881739976419156957473316, 4.20465437540909186227380024054, 5.32027457550671205414922003503, 5.76198737946871136007362888362, 7.05956508171912337153803989178, 8.241147477535537178114824731863, 9.070009141545368114370431480391, 10.09868768383671068710163975115, 11.48282574442425020667876943096, 12.50514349076677801374128331738

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