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Properties

Label 2-180-4.3-c8-0-21
Degree $2$
Conductor $180$
Sign $-0.984 + 0.175i$
Analytic cond. $73.3281$
Root an. cond. $8.56318$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.41 + 15.9i)2-s + (−252. + 45.0i)4-s − 279.·5-s + 2.63e3i·7-s + (−1.07e3 − 3.95e3i)8-s + (−395. − 4.45e3i)10-s + 2.48e3i·11-s + 4.13e4·13-s + (−4.19e4 + 3.72e3i)14-s + (6.14e4 − 2.27e4i)16-s + 4.86e4·17-s + 2.41e5i·19-s + (7.04e4 − 1.25e4i)20-s + (−3.95e4 + 3.51e3i)22-s − 5.77e4i·23-s + ⋯
L(s)  = 1  + (0.0883 + 0.996i)2-s + (−0.984 + 0.175i)4-s − 0.447·5-s + 1.09i·7-s + (−0.262 − 0.965i)8-s + (−0.0395 − 0.445i)10-s + 0.169i·11-s + 1.44·13-s + (−1.09 + 0.0968i)14-s + (0.938 − 0.346i)16-s + 0.582·17-s + 1.85i·19-s + (0.440 − 0.0786i)20-s + (−0.169 + 0.0149i)22-s − 0.206i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 + 0.175i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.984 + 0.175i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $-0.984 + 0.175i$
Analytic conductor: \(73.3281\)
Root analytic conductor: \(8.56318\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 180,\ (\ :4),\ -0.984 + 0.175i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.587857325\)
\(L(\frac12)\) \(\approx\) \(1.587857325\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 - 15.9i)T \)
3 \( 1 \)
5 \( 1 + 279.T \)
good7 \( 1 - 2.63e3iT - 5.76e6T^{2} \)
11 \( 1 - 2.48e3iT - 2.14e8T^{2} \)
13 \( 1 - 4.13e4T + 8.15e8T^{2} \)
17 \( 1 - 4.86e4T + 6.97e9T^{2} \)
19 \( 1 - 2.41e5iT - 1.69e10T^{2} \)
23 \( 1 + 5.77e4iT - 7.83e10T^{2} \)
29 \( 1 - 7.29e5T + 5.00e11T^{2} \)
31 \( 1 - 1.34e5iT - 8.52e11T^{2} \)
37 \( 1 - 1.68e6T + 3.51e12T^{2} \)
41 \( 1 - 7.61e5T + 7.98e12T^{2} \)
43 \( 1 + 2.54e6iT - 1.16e13T^{2} \)
47 \( 1 - 5.65e6iT - 2.38e13T^{2} \)
53 \( 1 + 1.14e7T + 6.22e13T^{2} \)
59 \( 1 - 2.12e7iT - 1.46e14T^{2} \)
61 \( 1 + 2.63e7T + 1.91e14T^{2} \)
67 \( 1 + 1.53e6iT - 4.06e14T^{2} \)
71 \( 1 + 2.06e7iT - 6.45e14T^{2} \)
73 \( 1 - 2.51e7T + 8.06e14T^{2} \)
79 \( 1 - 1.95e7iT - 1.51e15T^{2} \)
83 \( 1 + 2.37e7iT - 2.25e15T^{2} \)
89 \( 1 + 3.86e7T + 3.93e15T^{2} \)
97 \( 1 + 9.46e7T + 7.83e15T^{2} \)
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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

−12.02546951441590748101289428477, −10.58895217309938259371908651550, −9.365724562961191920753025836872, −8.426941465958314244947399364064, −7.77970733883147913467328857391, −6.28482663998513930201430268007, −5.69535632267995866098541578879, −4.31956774869405510119369124427, −3.19576742038977313374552835017, −1.24624234776046017523558607660, 0.45196485888797831235194949040, 1.23699349761680813562287752230, 2.95438079186517088023141435749, 3.90641399742160760929821007751, 4.86598836427616894979529886306, 6.41130223577031673889731340463, 7.78099401588009463932557989473, 8.765985625787882155236680925496, 9.847819205471135315640812248330, 10.98511540817060724461536044635

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