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Properties

Label 2-6e4-144.29-c0-0-1
Degree $2$
Conductor $1296$
Sign $0.976 - 0.216i$
Analytic cond. $0.646788$
Root an. cond. $0.804231$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.965 − 0.258i)2-s + (0.866 − 0.499i)4-s + (−0.258 + 0.965i)5-s + (0.866 + 0.5i)7-s + (0.707 − 0.707i)8-s + i·10-s + (−0.965 + 0.258i)11-s + (−0.366 + 1.36i)13-s + (0.965 + 0.258i)14-s + (0.500 − 0.866i)16-s − 1.41i·17-s + (0.258 + 0.965i)20-s + (−0.866 + 0.499i)22-s + 1.41i·26-s + 28-s + ⋯
L(s)  = 1  + (0.965 − 0.258i)2-s + (0.866 − 0.499i)4-s + (−0.258 + 0.965i)5-s + (0.866 + 0.5i)7-s + (0.707 − 0.707i)8-s + i·10-s + (−0.965 + 0.258i)11-s + (−0.366 + 1.36i)13-s + (0.965 + 0.258i)14-s + (0.500 − 0.866i)16-s − 1.41i·17-s + (0.258 + 0.965i)20-s + (−0.866 + 0.499i)22-s + 1.41i·26-s + 28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 - 0.216i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 - 0.216i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1296\)    =    \(2^{4} \cdot 3^{4}\)
Sign: $0.976 - 0.216i$
Analytic conductor: \(0.646788\)
Root analytic conductor: \(0.804231\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1296} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1296,\ (\ :0),\ 0.976 - 0.216i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.912877195\)
\(L(\frac12)\) \(\approx\) \(1.912877195\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 + 0.258i)T \)
3 \( 1 \)
good5 \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \)
7 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \)
13 \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \)
17 \( 1 + 1.41iT - T^{2} \)
19 \( 1 + iT^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.866 + 0.5i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \)
47 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
59 \( 1 + (-0.866 - 0.5i)T^{2} \)
61 \( 1 + (0.866 - 0.5i)T^{2} \)
67 \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \)
71 \( 1 - 1.41T + T^{2} \)
73 \( 1 - iT - T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2} \)
89 \( 1 + 1.41T + T^{2} \)
97 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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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

−10.07838394964023082931802188961, −9.254346425281483725846433194605, −7.993216446772210206501373143698, −7.16910999848109932751713066395, −6.68832269543377671206814845148, −5.39113785547304746867545729560, −4.88557001036998110086296712836, −3.82483152062545842652241371049, −2.66758290814143449587328425215, −2.03216965210974710039802786637, 1.46333307752909218992550674978, 2.89117696893927574312423034488, 3.95050538779211238528314418724, 4.92091984394789219007278782743, 5.29992101021058243909084336331, 6.32104658030097408435492958392, 7.55869747560813391438125119577, 8.084903358587861569480669274068, 8.537921870377032399430259036189, 10.11104662206574785398004058619

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