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Properties

Label 2-6e4-9.7-c1-0-19
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  + (−0.866 − 1.5i)5-s + (1 − 1.73i)7-s + (1.73 − 3i)11-s + (0.5 + 0.866i)13-s − 5.19·17-s − 2·19-s + (−1.73 − 3i)23-s + (1 − 1.73i)25-s + (0.866 − 1.5i)29-s + (4 + 6.92i)31-s − 3.46·35-s − 7·37-s + (−3.46 − 6i)41-s + (1 − 1.73i)43-s + (−3.46 + 6i)47-s + ⋯
L(s)  = 1  + (−0.387 − 0.670i)5-s + (0.377 − 0.654i)7-s + (0.522 − 0.904i)11-s + (0.138 + 0.240i)13-s − 1.26·17-s − 0.458·19-s + (−0.361 − 0.625i)23-s + (0.200 − 0.346i)25-s + (0.160 − 0.278i)29-s + (0.718 + 1.24i)31-s − 0.585·35-s − 1.15·37-s + (−0.541 − 0.937i)41-s + (0.152 − 0.264i)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

\(L(1)\) \(\approx\) \(1.114296870\)
\(L(\frac12)\) \(\approx\) \(1.114296870\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 + 1.5i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1 + 1.73i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.73 + 3i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 5.19T + 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 + (1.73 + 3i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.866 + 1.5i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4 - 6.92i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 7T + 37T^{2} \)
41 \( 1 + (3.46 + 6i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1 + 1.73i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.46 - 6i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + (6.92 + 12i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.5 + 6.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5 + 8.66i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 10.3T + 71T^{2} \)
73 \( 1 + 7T + 73T^{2} \)
79 \( 1 + (-1 + 1.73i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6.92 + 12i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 5.19T + 89T^{2} \)
97 \( 1 + (1 - 1.73i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.029701493158351292173214995664, −8.658928390754821985815511931123, −7.905214117753934477331941991732, −6.80005022913607574130046645892, −6.19507322753852152035948250691, −4.83707616380996170353489201261, −4.33500927875733169395439000441, −3.30058562508724548369826172602, −1.78493209059767483978624707023, −0.45792376923838425383809520082, 1.75477546705600944471162419176, 2.76091242841306581366912139298, 3.95046881043437173709420270115, 4.76982850492745526921500692138, 5.86898257940827926391675801933, 6.75967603040377352651697902008, 7.39780538412219946047757389071, 8.403982606156054018711126155094, 9.049223950364357535657091710043, 9.993345288018417177443065109892

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