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Properties

Label 2-180-12.11-c7-0-23
Degree $2$
Conductor $180$
Sign $0.576 - 0.817i$
Analytic cond. $56.2293$
Root an. cond. $7.49862$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.91 + 11.1i)2-s + (−120. + 42.7i)4-s + 125i·5-s − 691. i·7-s + (−708. − 1.26e3i)8-s + (−1.39e3 + 239. i)10-s − 5.08e3·11-s − 7.32e3·13-s + (7.70e3 − 1.32e3i)14-s + (1.27e4 − 1.03e4i)16-s − 6.75e3i·17-s + 4.65e4i·19-s + (−5.34e3 − 1.50e4i)20-s + (−9.75e3 − 5.67e4i)22-s + 1.03e5·23-s + ⋯
L(s)  = 1  + (0.169 + 0.985i)2-s + (−0.942 + 0.334i)4-s + 0.447i·5-s − 0.761i·7-s + (−0.489 − 0.872i)8-s + (−0.440 + 0.0758i)10-s − 1.15·11-s − 0.925·13-s + (0.750 − 0.129i)14-s + (0.776 − 0.629i)16-s − 0.333i·17-s + 1.55i·19-s + (−0.149 − 0.421i)20-s + (−0.195 − 1.13i)22-s + 1.76·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.576 - 0.817i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.576 - 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $0.576 - 0.817i$
Analytic conductor: \(56.2293\)
Root analytic conductor: \(7.49862\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 180,\ (\ :7/2),\ 0.576 - 0.817i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.481029200\)
\(L(\frac12)\) \(\approx\) \(1.481029200\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.91 - 11.1i)T \)
3 \( 1 \)
5 \( 1 - 125iT \)
good7 \( 1 + 691. iT - 8.23e5T^{2} \)
11 \( 1 + 5.08e3T + 1.94e7T^{2} \)
13 \( 1 + 7.32e3T + 6.27e7T^{2} \)
17 \( 1 + 6.75e3iT - 4.10e8T^{2} \)
19 \( 1 - 4.65e4iT - 8.93e8T^{2} \)
23 \( 1 - 1.03e5T + 3.40e9T^{2} \)
29 \( 1 + 2.14e5iT - 1.72e10T^{2} \)
31 \( 1 + 2.33e5iT - 2.75e10T^{2} \)
37 \( 1 - 2.03e5T + 9.49e10T^{2} \)
41 \( 1 - 2.36e5iT - 1.94e11T^{2} \)
43 \( 1 - 9.14e5iT - 2.71e11T^{2} \)
47 \( 1 + 7.71e4T + 5.06e11T^{2} \)
53 \( 1 + 6.08e5iT - 1.17e12T^{2} \)
59 \( 1 + 6.70e5T + 2.48e12T^{2} \)
61 \( 1 - 3.07e6T + 3.14e12T^{2} \)
67 \( 1 - 3.28e6iT - 6.06e12T^{2} \)
71 \( 1 + 1.57e6T + 9.09e12T^{2} \)
73 \( 1 - 2.99e6T + 1.10e13T^{2} \)
79 \( 1 + 5.55e6iT - 1.92e13T^{2} \)
83 \( 1 - 6.21e6T + 2.71e13T^{2} \)
89 \( 1 - 2.89e6iT - 4.42e13T^{2} \)
97 \( 1 - 4.14e6T + 8.07e13T^{2} \)
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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

−11.54021212615924723174689929732, −10.24667689914353264202073897467, −9.568175569319779302471356270584, −7.945893593390007216154821802482, −7.53698773237309212741175857139, −6.35108920030642235393111373673, −5.21350544520861593598492335799, −4.11396247205136910091809788360, −2.73925202836039257062137553538, −0.57242275136444377893482773419, 0.70503225679796811844652117382, 2.23523452191576885232792323669, 3.12474793386481959336130131373, 4.92654477101328801464109649990, 5.26703703458941874195621184046, 7.12816503908759105965586724298, 8.638882342906976909234382700829, 9.164950883872409665003758072181, 10.41997700218245453304911124923, 11.17203382902630009386560564343

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