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Properties

Label 2-180-12.11-c5-0-17
Degree $2$
Conductor $180$
Sign $-0.193 - 0.981i$
Analytic cond. $28.8690$
Root an. cond. $5.37299$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.07 + 4.74i)2-s + (−13.0 + 29.2i)4-s − 25i·5-s − 82.8i·7-s + (−178. + 27.9i)8-s + (118. − 76.9i)10-s + 461.·11-s + 520.·13-s + (393. − 255. i)14-s + (−682. − 763. i)16-s + 2.04e3i·17-s + 1.90e3i·19-s + (730. + 326. i)20-s + (1.42e3 + 2.19e3i)22-s + 973.·23-s + ⋯
L(s)  = 1  + (0.544 + 0.839i)2-s + (−0.408 + 0.912i)4-s − 0.447i·5-s − 0.639i·7-s + (−0.988 + 0.154i)8-s + (0.375 − 0.243i)10-s + 1.15·11-s + 0.854·13-s + (0.536 − 0.347i)14-s + (−0.666 − 0.745i)16-s + 1.72i·17-s + 1.21i·19-s + (0.408 + 0.182i)20-s + (0.625 + 0.965i)22-s + 0.383·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.193 - 0.981i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.193 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $-0.193 - 0.981i$
Analytic conductor: \(28.8690\)
Root analytic conductor: \(5.37299\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 180,\ (\ :5/2),\ -0.193 - 0.981i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.541023988\)
\(L(\frac12)\) \(\approx\) \(2.541023988\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-3.07 - 4.74i)T \)
3 \( 1 \)
5 \( 1 + 25iT \)
good7 \( 1 + 82.8iT - 1.68e4T^{2} \)
11 \( 1 - 461.T + 1.61e5T^{2} \)
13 \( 1 - 520.T + 3.71e5T^{2} \)
17 \( 1 - 2.04e3iT - 1.41e6T^{2} \)
19 \( 1 - 1.90e3iT - 2.47e6T^{2} \)
23 \( 1 - 973.T + 6.43e6T^{2} \)
29 \( 1 + 6.65e3iT - 2.05e7T^{2} \)
31 \( 1 - 8.81e3iT - 2.86e7T^{2} \)
37 \( 1 - 6.27e3T + 6.93e7T^{2} \)
41 \( 1 - 1.09e4iT - 1.15e8T^{2} \)
43 \( 1 - 8.33e3iT - 1.47e8T^{2} \)
47 \( 1 - 850.T + 2.29e8T^{2} \)
53 \( 1 + 1.65e4iT - 4.18e8T^{2} \)
59 \( 1 + 3.70e4T + 7.14e8T^{2} \)
61 \( 1 - 1.32e4T + 8.44e8T^{2} \)
67 \( 1 - 5.37e4iT - 1.35e9T^{2} \)
71 \( 1 - 5.34e4T + 1.80e9T^{2} \)
73 \( 1 - 2.80e4T + 2.07e9T^{2} \)
79 \( 1 - 5.45e4iT - 3.07e9T^{2} \)
83 \( 1 + 6.17e4T + 3.93e9T^{2} \)
89 \( 1 + 1.95e4iT - 5.58e9T^{2} \)
97 \( 1 - 7.64e4T + 8.58e9T^{2} \)
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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

−12.34447396136373063149891834584, −11.25224333139188719802937987461, −9.909104731906418712061710963357, −8.662402713238518033129991442610, −7.971963754921984804468207908316, −6.59920774789512017950466798433, −5.87214350760215102235806053032, −4.31894674139771062162441911699, −3.62872648521951957328727047873, −1.29625852908245585507706696941, 0.77911025408449917503768485608, 2.35658149597796969077684334315, 3.47852375052775697553016874576, 4.80071397775422374231828429240, 6.00547507223239780265340659427, 7.06784634982450792627324578918, 8.979058656718839012661833035526, 9.397322093042104810715721297505, 10.87915180895051323662819142405, 11.47458990027955123152114684314

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