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Properties

Label 2-360-8.5-c3-0-7
Degree $2$
Conductor $360$
Sign $-0.961 - 0.274i$
Analytic cond. $21.2406$
Root an. cond. $4.60876$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.57 + 1.18i)2-s + (5.21 − 6.07i)4-s + 5i·5-s + 1.05·7-s + (−6.22 + 21.7i)8-s + (−5.90 − 12.8i)10-s + 12.0i·11-s + 1.58i·13-s + (−2.72 + 1.25i)14-s + (−9.69 − 63.2i)16-s + 5.63·17-s + 26.2i·19-s + (30.3 + 26.0i)20-s + (−14.2 − 30.9i)22-s − 104.·23-s + ⋯
L(s)  = 1  + (−0.908 + 0.417i)2-s + (0.651 − 0.758i)4-s + 0.447i·5-s + 0.0571·7-s + (−0.274 + 0.961i)8-s + (−0.186 − 0.406i)10-s + 0.329i·11-s + 0.0337i·13-s + (−0.0519 + 0.0238i)14-s + (−0.151 − 0.988i)16-s + 0.0803·17-s + 0.317i·19-s + (0.339 + 0.291i)20-s + (−0.137 − 0.299i)22-s − 0.944·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.961 - 0.274i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.961 - 0.274i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $-0.961 - 0.274i$
Analytic conductor: \(21.2406\)
Root analytic conductor: \(4.60876\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :3/2),\ -0.961 - 0.274i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.5447898769\)
\(L(\frac12)\) \(\approx\) \(0.5447898769\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.57 - 1.18i)T \)
3 \( 1 \)
5 \( 1 - 5iT \)
good7 \( 1 - 1.05T + 343T^{2} \)
11 \( 1 - 12.0iT - 1.33e3T^{2} \)
13 \( 1 - 1.58iT - 2.19e3T^{2} \)
17 \( 1 - 5.63T + 4.91e3T^{2} \)
19 \( 1 - 26.2iT - 6.85e3T^{2} \)
23 \( 1 + 104.T + 1.21e4T^{2} \)
29 \( 1 + 32.7iT - 2.43e4T^{2} \)
31 \( 1 - 250.T + 2.97e4T^{2} \)
37 \( 1 - 235. iT - 5.06e4T^{2} \)
41 \( 1 + 138.T + 6.89e4T^{2} \)
43 \( 1 - 355. iT - 7.95e4T^{2} \)
47 \( 1 + 462.T + 1.03e5T^{2} \)
53 \( 1 + 5.63iT - 1.48e5T^{2} \)
59 \( 1 + 46.7iT - 2.05e5T^{2} \)
61 \( 1 + 340. iT - 2.26e5T^{2} \)
67 \( 1 - 790. iT - 3.00e5T^{2} \)
71 \( 1 + 971.T + 3.57e5T^{2} \)
73 \( 1 + 1.02e3T + 3.89e5T^{2} \)
79 \( 1 + 693.T + 4.93e5T^{2} \)
83 \( 1 + 117. iT - 5.71e5T^{2} \)
89 \( 1 + 106.T + 7.04e5T^{2} \)
97 \( 1 + 1.26e3T + 9.12e5T^{2} \)
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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

−11.34035430726936800443901677676, −10.14116467657909546071327237239, −9.812513057478544940181014364879, −8.485758207108166814478968078262, −7.81396829509060883295615479565, −6.74016144784824281371560335604, −5.98350260653171179312687484980, −4.63956562593909087622716997036, −2.91264814314145248419432490598, −1.52504411661233307813842971202, 0.25543250315042027214580239153, 1.67942588962514300308361582134, 3.07438562215999572956773134112, 4.37467290882880849624525786073, 5.86300835032730839399688617233, 6.99097465847656014553074725663, 8.064335642568219521749564938652, 8.724578750765811796380696504219, 9.706263319831387615288731480540, 10.46739251737382310825673647482

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