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Properties

Label 2-180-4.3-c6-0-0
Degree $2$
Conductor $180$
Sign $0.630 - 0.776i$
Analytic cond. $41.4097$
Root an. cond. $6.43503$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.43 − 7.22i)2-s + (−40.3 + 49.6i)4-s − 55.9·5-s − 472. i·7-s + (497. + 120. i)8-s + (192. + 403. i)10-s + 939. i·11-s + 22.5·13-s + (−3.40e3 + 1.62e3i)14-s + (−839. − 4.00e3i)16-s − 6.24e3·17-s − 1.11e4i·19-s + (2.25e3 − 2.77e3i)20-s + (6.78e3 − 3.23e3i)22-s − 1.16e3i·23-s + ⋯
L(s)  = 1  + (−0.429 − 0.902i)2-s + (−0.630 + 0.776i)4-s − 0.447·5-s − 1.37i·7-s + (0.971 + 0.235i)8-s + (0.192 + 0.403i)10-s + 0.705i·11-s + 0.0102·13-s + (−1.24 + 0.591i)14-s + (−0.205 − 0.978i)16-s − 1.27·17-s − 1.62i·19-s + (0.281 − 0.347i)20-s + (0.637 − 0.303i)22-s − 0.0960i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.630 - 0.776i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.630 - 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $0.630 - 0.776i$
Analytic conductor: \(41.4097\)
Root analytic conductor: \(6.43503\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 180,\ (\ :3),\ 0.630 - 0.776i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.3656016983\)
\(L(\frac12)\) \(\approx\) \(0.3656016983\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (3.43 + 7.22i)T \)
3 \( 1 \)
5 \( 1 + 55.9T \)
good7 \( 1 + 472. iT - 1.17e5T^{2} \)
11 \( 1 - 939. iT - 1.77e6T^{2} \)
13 \( 1 - 22.5T + 4.82e6T^{2} \)
17 \( 1 + 6.24e3T + 2.41e7T^{2} \)
19 \( 1 + 1.11e4iT - 4.70e7T^{2} \)
23 \( 1 + 1.16e3iT - 1.48e8T^{2} \)
29 \( 1 + 2.56e4T + 5.94e8T^{2} \)
31 \( 1 + 4.30e3iT - 8.87e8T^{2} \)
37 \( 1 + 5.06e4T + 2.56e9T^{2} \)
41 \( 1 - 1.13e5T + 4.75e9T^{2} \)
43 \( 1 - 1.13e5iT - 6.32e9T^{2} \)
47 \( 1 - 1.84e5iT - 1.07e10T^{2} \)
53 \( 1 - 1.33e5T + 2.21e10T^{2} \)
59 \( 1 + 5.52e4iT - 4.21e10T^{2} \)
61 \( 1 - 2.04e5T + 5.15e10T^{2} \)
67 \( 1 - 2.44e5iT - 9.04e10T^{2} \)
71 \( 1 + 1.42e4iT - 1.28e11T^{2} \)
73 \( 1 + 2.78e5T + 1.51e11T^{2} \)
79 \( 1 - 3.58e5iT - 2.43e11T^{2} \)
83 \( 1 - 8.20e5iT - 3.26e11T^{2} \)
89 \( 1 + 1.00e6T + 4.96e11T^{2} \)
97 \( 1 - 1.44e6T + 8.32e11T^{2} \)
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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

−11.28254847539002447570795113457, −10.95693508264523984631401458357, −9.799051203535193492600748628553, −8.892431925568980946077667173948, −7.61218634432966432490610109475, −6.92902116497908783839567981507, −4.66837150750630087869157144364, −3.99620580186261379591223223186, −2.52124834942112688375216245065, −0.990999642317884579959549750430, 0.14486927022774846394052201737, 1.97939157831023620546153487287, 3.83832833255452083932068937602, 5.35728003717597590779508759746, 6.08784415615783368781238671875, 7.34635175431659836003656654538, 8.543193444312620450782356452644, 8.938137647142670313765199134649, 10.25870538939042213259708839923, 11.37365435631776825017394307671

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