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Properties

Label 2-180-4.3-c6-0-28
Degree $2$
Conductor $180$
Sign $0.702 - 0.711i$
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  + (3.08 + 7.38i)2-s + (−44.9 + 45.5i)4-s − 55.9·5-s − 200. i·7-s + (−474. − 191. i)8-s + (−172. − 412. i)10-s − 76.4i·11-s − 2.21e3·13-s + (1.47e3 − 616. i)14-s + (−51.2 − 4.09e3i)16-s + 7.76e3·17-s + 3.20e3i·19-s + (2.51e3 − 2.54e3i)20-s + (564. − 235. i)22-s − 1.24e4i·23-s + ⋯
L(s)  = 1  + (0.385 + 0.922i)2-s + (−0.702 + 0.711i)4-s − 0.447·5-s − 0.583i·7-s + (−0.927 − 0.373i)8-s + (−0.172 − 0.412i)10-s − 0.0574i·11-s − 1.00·13-s + (0.538 − 0.224i)14-s + (−0.0125 − 0.999i)16-s + 1.58·17-s + 0.467i·19-s + (0.314 − 0.318i)20-s + (0.0530 − 0.0221i)22-s − 1.02i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.702 - 0.711i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.702 - 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $0.702 - 0.711i$
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: \((2,\ 180,\ (\ :3),\ 0.702 - 0.711i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.831525996\)
\(L(\frac12)\) \(\approx\) \(1.831525996\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-3.08 - 7.38i)T \)
3 \( 1 \)
5 \( 1 + 55.9T \)
good7 \( 1 + 200. iT - 1.17e5T^{2} \)
11 \( 1 + 76.4iT - 1.77e6T^{2} \)
13 \( 1 + 2.21e3T + 4.82e6T^{2} \)
17 \( 1 - 7.76e3T + 2.41e7T^{2} \)
19 \( 1 - 3.20e3iT - 4.70e7T^{2} \)
23 \( 1 + 1.24e4iT - 1.48e8T^{2} \)
29 \( 1 - 3.66e4T + 5.94e8T^{2} \)
31 \( 1 - 1.37e4iT - 8.87e8T^{2} \)
37 \( 1 + 4.40e4T + 2.56e9T^{2} \)
41 \( 1 - 7.55e4T + 4.75e9T^{2} \)
43 \( 1 + 2.22e4iT - 6.32e9T^{2} \)
47 \( 1 - 1.76e5iT - 1.07e10T^{2} \)
53 \( 1 - 7.50e4T + 2.21e10T^{2} \)
59 \( 1 + 1.76e5iT - 4.21e10T^{2} \)
61 \( 1 + 3.26e4T + 5.15e10T^{2} \)
67 \( 1 + 3.09e5iT - 9.04e10T^{2} \)
71 \( 1 + 3.77e5iT - 1.28e11T^{2} \)
73 \( 1 - 4.23e5T + 1.51e11T^{2} \)
79 \( 1 - 6.43e5iT - 2.43e11T^{2} \)
83 \( 1 - 8.24e5iT - 3.26e11T^{2} \)
89 \( 1 - 5.39e5T + 4.96e11T^{2} \)
97 \( 1 - 1.03e6T + 8.32e11T^{2} \)
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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

−12.22258372485745464028420145024, −10.58700088402102000557371833592, −9.576887589180779599110945752466, −8.250528754511651002563065541571, −7.54217453337729515207966355464, −6.54863082026729106730821869030, −5.24741740186257258247713056558, −4.22599596656460540410257048334, −3.01749641254962833808240609233, −0.66505970118233584692957381583, 0.843268443614291036180401981736, 2.39092295212086891208839556021, 3.48997233029032248353777948954, 4.83041845206006744766420947734, 5.75594897038767013271350181263, 7.36502473605428993261103564017, 8.613304872036644195147473950680, 9.677136485449143864115415812903, 10.46705313461729494479659182461, 11.89400463909266325410594444859

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