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Properties

Label 2-6e4-9.7-c1-0-4
Degree $2$
Conductor $1296$
Sign $0.642 - 0.766i$
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  + (−0.133 − 0.232i)5-s + (−1.73 + 3i)7-s + (1 − 1.73i)11-s + (−2.23 − 3.86i)13-s + 5.73·17-s + 0.535·19-s + (4.46 + 7.73i)23-s + (2.46 − 4.26i)25-s + (−3.86 + 6.69i)29-s + (1.46 + 2.53i)31-s + 0.928·35-s + 6.46·37-s + (3.46 + 6i)41-s + (−5.73 + 9.92i)43-s + (−3.46 + 6i)47-s + ⋯
L(s)  = 1  + (−0.0599 − 0.103i)5-s + (−0.654 + 1.13i)7-s + (0.301 − 0.522i)11-s + (−0.619 − 1.07i)13-s + 1.39·17-s + 0.122·19-s + (0.930 + 1.61i)23-s + (0.492 − 0.853i)25-s + (−0.717 + 1.24i)29-s + (0.262 + 0.455i)31-s + 0.156·35-s + 1.06·37-s + (0.541 + 0.937i)41-s + (−0.874 + 1.51i)43-s + (−0.505 + 0.875i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1296\)    =    \(2^{4} \cdot 3^{4}\)
Sign: $0.642 - 0.766i$
Analytic conductor: \(10.3486\)
Root analytic conductor: \(3.21692\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1296} (865, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1296,\ (\ :1/2),\ 0.642 - 0.766i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.453069824\)
\(L(\frac12)\) \(\approx\) \(1.453069824\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.133 + 0.232i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (1.73 - 3i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.23 + 3.86i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 5.73T + 17T^{2} \)
19 \( 1 - 0.535T + 19T^{2} \)
23 \( 1 + (-4.46 - 7.73i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.86 - 6.69i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.46 - 2.53i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 6.46T + 37T^{2} \)
41 \( 1 + (-3.46 - 6i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.73 - 9.92i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.46 - 6i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 2.92T + 53T^{2} \)
59 \( 1 + (-4 - 6.92i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.76 - 3.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.73 + 6.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 - T + 73T^{2} \)
79 \( 1 + (-3.73 + 6.46i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.46 + 9.46i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 5.19T + 89T^{2} \)
97 \( 1 + (7.92 - 13.7i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.600133427746692284235428445445, −9.121569885257013069950932043733, −8.115264220601377585804209055455, −7.45872824352810743942837230828, −6.27902734788751085439838729003, −5.61416894389426054563401238099, −4.89490214262327639825602433334, −3.26427395944635024535497664185, −2.92566656476838702313046608316, −1.17913491424705456327558523189, 0.71649125260436449562158267899, 2.25101994968201484434343930837, 3.54034997707152686528211269438, 4.26003703005353174603330980385, 5.25013621643057795609022458633, 6.49085584484620274416402333391, 7.06812131941510082332099755290, 7.69396361465133549687382826965, 8.856911115910210579696808690284, 9.735210344022114704146944125784

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