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Properties

Label 2-180-12.11-c5-0-25
Degree $2$
Conductor $180$
Sign $0.336 + 0.941i$
Analytic cond. $28.8690$
Root an. cond. $5.37299$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.70 + 5.39i)2-s + (−26.1 − 18.3i)4-s + 25i·5-s + 101. i·7-s + (143. − 110. i)8-s + (−134. − 42.5i)10-s − 133.·11-s − 368.·13-s + (−546. − 172. i)14-s + (348. + 962. i)16-s − 415. i·17-s + 2.52e3i·19-s + (459. − 654. i)20-s + (227. − 721. i)22-s − 3.09e3·23-s + ⋯
L(s)  = 1  + (−0.301 + 0.953i)2-s + (−0.818 − 0.574i)4-s + 0.447i·5-s + 0.781i·7-s + (0.794 − 0.607i)8-s + (−0.426 − 0.134i)10-s − 0.333·11-s − 0.604·13-s + (−0.745 − 0.235i)14-s + (0.340 + 0.940i)16-s − 0.348i·17-s + 1.60i·19-s + (0.256 − 0.366i)20-s + (0.100 − 0.317i)22-s − 1.22·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.336 + 0.941i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.336 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $0.336 + 0.941i$
Analytic conductor: \(28.8690\)
Root analytic conductor: \(5.37299\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 180,\ (\ :5/2),\ 0.336 + 0.941i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.1904845587\)
\(L(\frac12)\) \(\approx\) \(0.1904845587\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.70 - 5.39i)T \)
3 \( 1 \)
5 \( 1 - 25iT \)
good7 \( 1 - 101. iT - 1.68e4T^{2} \)
11 \( 1 + 133.T + 1.61e5T^{2} \)
13 \( 1 + 368.T + 3.71e5T^{2} \)
17 \( 1 + 415. iT - 1.41e6T^{2} \)
19 \( 1 - 2.52e3iT - 2.47e6T^{2} \)
23 \( 1 + 3.09e3T + 6.43e6T^{2} \)
29 \( 1 + 3.65e3iT - 2.05e7T^{2} \)
31 \( 1 + 9.13e3iT - 2.86e7T^{2} \)
37 \( 1 - 3.57e3T + 6.93e7T^{2} \)
41 \( 1 + 1.29e4iT - 1.15e8T^{2} \)
43 \( 1 - 3.92e3iT - 1.47e8T^{2} \)
47 \( 1 + 7.43e3T + 2.29e8T^{2} \)
53 \( 1 + 2.43e4iT - 4.18e8T^{2} \)
59 \( 1 + 3.80e3T + 7.14e8T^{2} \)
61 \( 1 - 2.46e4T + 8.44e8T^{2} \)
67 \( 1 + 3.91e4iT - 1.35e9T^{2} \)
71 \( 1 - 1.00e4T + 1.80e9T^{2} \)
73 \( 1 + 2.18e4T + 2.07e9T^{2} \)
79 \( 1 - 2.49e4iT - 3.07e9T^{2} \)
83 \( 1 + 4.70e4T + 3.93e9T^{2} \)
89 \( 1 + 9.52e4iT - 5.58e9T^{2} \)
97 \( 1 + 1.43e5T + 8.58e9T^{2} \)
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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

−11.60281774202190222118488227013, −10.18045932350135619554523300928, −9.577573922531502205862194620534, −8.259743525358619578600734233899, −7.56144328443652978216286111971, −6.22299753568513373421243330256, −5.48005947811268146537229444887, −4.00817376474531328083210319326, −2.15605163745808579790720796901, −0.07149929498400167257969062102, 1.23947415053993523015277853053, 2.75432859232663676946965607364, 4.16435542913432767832986850244, 5.12701617606278602029897112825, 7.00988716039582972601229683492, 8.103883165146562728643451478758, 9.104158034213669046265296024573, 10.11077819604592922367756388461, 10.87897047185490958962252649990, 11.92733450813507225246857329346

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