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Properties

Label 2-6e4-12.11-c1-0-7
Degree $2$
Conductor $1296$
Sign $0.5 - 0.866i$
Analytic cond. $10.3486$
Root an. cond. $3.21692$
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  + 0.896i·5-s + 1.26i·7-s + 4.24·11-s + 13-s − 0.896i·17-s + 4.73i·19-s − 7.34·23-s + 4.19·25-s − 8.24i·29-s + 6i·31-s − 1.13·35-s + 1.19·37-s + 7.34i·41-s + 8.19i·43-s + 11.5·47-s + ⋯
L(s)  = 1  + 0.400i·5-s + 0.479i·7-s + 1.27·11-s + 0.277·13-s − 0.217i·17-s + 1.08i·19-s − 1.53·23-s + 0.839·25-s − 1.53i·29-s + 1.07i·31-s − 0.192·35-s + 0.196·37-s + 1.14i·41-s + 1.24i·43-s + 1.69·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.5 - 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.5 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1296\)    =    \(2^{4} \cdot 3^{4}\)
Sign: $0.5 - 0.866i$
Analytic conductor: \(10.3486\)
Root analytic conductor: \(3.21692\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1296} (1295, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1296,\ (\ :1/2),\ 0.5 - 0.866i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.694170041\)
\(L(\frac12)\) \(\approx\) \(1.694170041\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 - 0.896iT - 5T^{2} \)
7 \( 1 - 1.26iT - 7T^{2} \)
11 \( 1 - 4.24T + 11T^{2} \)
13 \( 1 - T + 13T^{2} \)
17 \( 1 + 0.896iT - 17T^{2} \)
19 \( 1 - 4.73iT - 19T^{2} \)
23 \( 1 + 7.34T + 23T^{2} \)
29 \( 1 + 8.24iT - 29T^{2} \)
31 \( 1 - 6iT - 31T^{2} \)
37 \( 1 - 1.19T + 37T^{2} \)
41 \( 1 - 7.34iT - 41T^{2} \)
43 \( 1 - 8.19iT - 43T^{2} \)
47 \( 1 - 11.5T + 47T^{2} \)
53 \( 1 - 7.34iT - 53T^{2} \)
59 \( 1 + 11.5T + 59T^{2} \)
61 \( 1 + 3.19T + 61T^{2} \)
67 \( 1 - 10.7iT - 67T^{2} \)
71 \( 1 + 1.13T + 71T^{2} \)
73 \( 1 - 9.19T + 73T^{2} \)
79 \( 1 - 4.73iT - 79T^{2} \)
83 \( 1 - 8.48T + 83T^{2} \)
89 \( 1 + 11.8iT - 89T^{2} \)
97 \( 1 - 14.3T + 97T^{2} \)
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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

−9.764224513122187863726904825383, −9.015738508367990125306509601240, −8.199557240038105856915522505289, −7.36660640027850614021625305118, −6.24065664411591862747210648974, −5.96810644843818381620141979250, −4.53472878403604257458771760505, −3.73745749403307057825780352593, −2.61656845367979174074258994556, −1.36276344377961105051424729666, 0.795097190894943578618211415835, 2.07089597361678431584136503355, 3.58130597224761691897949914836, 4.24782452120333598159903714026, 5.27281751486633025704006825540, 6.30502807720138724822485326936, 7.00526053557312362286593706753, 7.894488965268906723215448134539, 9.000098971278985353422854655172, 9.193832041344784887492483858375

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