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Properties

Label 2-180-4.3-c6-0-17
Degree $2$
Conductor $180$
Sign $-0.0871 - 0.996i$
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  + (−5.40 + 5.89i)2-s + (−5.57 − 63.7i)4-s + 55.9·5-s − 125. i·7-s + (406. + 311. i)8-s + (−302. + 329. i)10-s + 685. i·11-s − 3.31e3·13-s + (739. + 677. i)14-s + (−4.03e3 + 711. i)16-s + 3.07e3·17-s − 6.08e3i·19-s + (−311. − 3.56e3i)20-s + (−4.04e3 − 3.70e3i)22-s + 6.97e3i·23-s + ⋯
L(s)  = 1  + (−0.675 + 0.737i)2-s + (−0.0871 − 0.996i)4-s + 0.447·5-s − 0.365i·7-s + (0.793 + 0.608i)8-s + (−0.302 + 0.329i)10-s + 0.515i·11-s − 1.50·13-s + (0.269 + 0.247i)14-s + (−0.984 + 0.173i)16-s + 0.625·17-s − 0.886i·19-s + (−0.0389 − 0.445i)20-s + (−0.379 − 0.348i)22-s + 0.573i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0871 - 0.996i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.0871 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $-0.0871 - 0.996i$
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: \((2,\ 180,\ (\ :3),\ -0.0871 - 0.996i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.173836283\)
\(L(\frac12)\) \(\approx\) \(1.173836283\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (5.40 - 5.89i)T \)
3 \( 1 \)
5 \( 1 - 55.9T \)
good7 \( 1 + 125. iT - 1.17e5T^{2} \)
11 \( 1 - 685. iT - 1.77e6T^{2} \)
13 \( 1 + 3.31e3T + 4.82e6T^{2} \)
17 \( 1 - 3.07e3T + 2.41e7T^{2} \)
19 \( 1 + 6.08e3iT - 4.70e7T^{2} \)
23 \( 1 - 6.97e3iT - 1.48e8T^{2} \)
29 \( 1 - 1.37e4T + 5.94e8T^{2} \)
31 \( 1 - 3.75e4iT - 8.87e8T^{2} \)
37 \( 1 - 6.43e4T + 2.56e9T^{2} \)
41 \( 1 + 2.74e3T + 4.75e9T^{2} \)
43 \( 1 + 1.00e5iT - 6.32e9T^{2} \)
47 \( 1 - 7.29e4iT - 1.07e10T^{2} \)
53 \( 1 + 9.96e4T + 2.21e10T^{2} \)
59 \( 1 - 2.37e5iT - 4.21e10T^{2} \)
61 \( 1 - 2.74e5T + 5.15e10T^{2} \)
67 \( 1 - 4.44e5iT - 9.04e10T^{2} \)
71 \( 1 - 3.24e5iT - 1.28e11T^{2} \)
73 \( 1 + 6.89e5T + 1.51e11T^{2} \)
79 \( 1 - 1.47e5iT - 2.43e11T^{2} \)
83 \( 1 - 3.55e5iT - 3.26e11T^{2} \)
89 \( 1 + 2.48e4T + 4.96e11T^{2} \)
97 \( 1 + 9.38e5T + 8.32e11T^{2} \)
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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

−11.70313315496861164898536104235, −10.38440637777459521131124786454, −9.804219684412744229194356884155, −8.839838158369982932719773742483, −7.52058527238490868316976323077, −6.91938202041444416727047962536, −5.54689168708683760602934751776, −4.59525261886604632921658320760, −2.47295729645329488459519129359, −0.996709525913453008983179158453, 0.50062636022561280164179625631, 2.01482296246582127219829097366, 3.05814929860142900724104912558, 4.59706702851241549193468338745, 6.03696647549458413326202643281, 7.47503388021219373406009459959, 8.371618435296910024781176121445, 9.577930204012268301229110673031, 10.08288413356415951953187017625, 11.27933872085680048825105165409

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