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Properties

Label 2-240-15.8-c3-0-3
Degree $2$
Conductor $240$
Sign $-0.846 + 0.533i$
Analytic cond. $14.1604$
Root an. cond. $3.76303$
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  + (2.80 + 4.37i)3-s + (−9.55 + 5.80i)5-s + (−9.35 + 9.35i)7-s + (−11.2 + 24.5i)9-s − 34.1i·11-s + (2.82 + 2.82i)13-s + (−52.1 − 25.5i)15-s + (−64.2 − 64.2i)17-s − 19.0i·19-s + (−67.1 − 14.6i)21-s + (−51.4 + 51.4i)23-s + (57.6 − 110. i)25-s + (−138. + 19.5i)27-s − 50.5·29-s + 93.3·31-s + ⋯
L(s)  = 1  + (0.539 + 0.841i)3-s + (−0.854 + 0.518i)5-s + (−0.505 + 0.505i)7-s + (−0.417 + 0.908i)9-s − 0.935i·11-s + (0.0601 + 0.0601i)13-s + (−0.898 − 0.439i)15-s + (−0.916 − 0.916i)17-s − 0.230i·19-s + (−0.698 − 0.152i)21-s + (−0.466 + 0.466i)23-s + (0.461 − 0.887i)25-s + (−0.990 + 0.139i)27-s − 0.323·29-s + 0.540·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.846 + 0.533i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.846 + 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $-0.846 + 0.533i$
Analytic conductor: \(14.1604\)
Root analytic conductor: \(3.76303\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :3/2),\ -0.846 + 0.533i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.108269 - 0.374994i\)
\(L(\frac12)\) \(\approx\) \(0.108269 - 0.374994i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-2.80 - 4.37i)T \)
5 \( 1 + (9.55 - 5.80i)T \)
good7 \( 1 + (9.35 - 9.35i)T - 343iT^{2} \)
11 \( 1 + 34.1iT - 1.33e3T^{2} \)
13 \( 1 + (-2.82 - 2.82i)T + 2.19e3iT^{2} \)
17 \( 1 + (64.2 + 64.2i)T + 4.91e3iT^{2} \)
19 \( 1 + 19.0iT - 6.85e3T^{2} \)
23 \( 1 + (51.4 - 51.4i)T - 1.21e4iT^{2} \)
29 \( 1 + 50.5T + 2.43e4T^{2} \)
31 \( 1 - 93.3T + 2.97e4T^{2} \)
37 \( 1 + (161. - 161. i)T - 5.06e4iT^{2} \)
41 \( 1 + 88.7iT - 6.89e4T^{2} \)
43 \( 1 + (176. + 176. i)T + 7.95e4iT^{2} \)
47 \( 1 + (-38.2 - 38.2i)T + 1.03e5iT^{2} \)
53 \( 1 + (344. - 344. i)T - 1.48e5iT^{2} \)
59 \( 1 + 421.T + 2.05e5T^{2} \)
61 \( 1 - 2T + 2.26e5T^{2} \)
67 \( 1 + (430. - 430. i)T - 3.00e5iT^{2} \)
71 \( 1 - 733. iT - 3.57e5T^{2} \)
73 \( 1 + (348. + 348. i)T + 3.89e5iT^{2} \)
79 \( 1 + 588. iT - 4.93e5T^{2} \)
83 \( 1 + (217. - 217. i)T - 5.71e5iT^{2} \)
89 \( 1 - 1.27e3T + 7.04e5T^{2} \)
97 \( 1 + (432. - 432. i)T - 9.12e5iT^{2} \)
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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

−11.97182194426666159065991254611, −11.23479890537035384434768980531, −10.37434101937982791713953354467, −9.232485706475871021415956245749, −8.524130676790927568274399685859, −7.43372993524379621706076574059, −6.15304525061696230929758709285, −4.76793102033506530141068625798, −3.55751966310368113278561957556, −2.68918558388169765350920666714, 0.14049347735738941159822298910, 1.78956877334809452698195534715, 3.46455778223845762985193954524, 4.52368880186794935241966920487, 6.29594471089938549139819283983, 7.21560772666370634031637443450, 8.083607462835106818244368278434, 8.939253377700219323622936835593, 10.06872438937009934634419437034, 11.31413635807014090880803654941

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