Properties

Label 2-2160-5.4-c1-0-22
Degree $2$
Conductor $2160$
Sign $0.316 - 0.948i$
Analytic cond. $17.2476$
Root an. cond. $4.15303$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.12 + 0.707i)5-s + 3i·7-s + 4.24·11-s + 3i·13-s − 2.82i·17-s + 19-s + 7.07i·23-s + (3.99 + 3i)25-s − 4.24·29-s − 2·31-s + (−2.12 + 6.36i)35-s − 9i·37-s + 4.24·41-s − 6i·43-s + 2.82i·47-s + ⋯
L(s)  = 1  + (0.948 + 0.316i)5-s + 1.13i·7-s + 1.27·11-s + 0.832i·13-s − 0.685i·17-s + 0.229·19-s + 1.47i·23-s + (0.799 + 0.600i)25-s − 0.787·29-s − 0.359·31-s + (−0.358 + 1.07i)35-s − 1.47i·37-s + 0.662·41-s − 0.914i·43-s + 0.412i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.316 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.316 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2160\)    =    \(2^{4} \cdot 3^{3} \cdot 5\)
Sign: $0.316 - 0.948i$
Analytic conductor: \(17.2476\)
Root analytic conductor: \(4.15303\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2160} (1729, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2160,\ (\ :1/2),\ 0.316 - 0.948i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.244131301\)
\(L(\frac12)\) \(\approx\) \(2.244131301\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + (-2.12 - 0.707i)T \)
good7 \( 1 - 3iT - 7T^{2} \)
11 \( 1 - 4.24T + 11T^{2} \)
13 \( 1 - 3iT - 13T^{2} \)
17 \( 1 + 2.82iT - 17T^{2} \)
19 \( 1 - T + 19T^{2} \)
23 \( 1 - 7.07iT - 23T^{2} \)
29 \( 1 + 4.24T + 29T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 + 9iT - 37T^{2} \)
41 \( 1 - 4.24T + 41T^{2} \)
43 \( 1 + 6iT - 43T^{2} \)
47 \( 1 - 2.82iT - 47T^{2} \)
53 \( 1 - 9.89iT - 53T^{2} \)
59 \( 1 - 8.48T + 59T^{2} \)
61 \( 1 + 13T + 61T^{2} \)
67 \( 1 + 3iT - 67T^{2} \)
71 \( 1 + 12.7T + 71T^{2} \)
73 \( 1 - 9iT - 73T^{2} \)
79 \( 1 + 5T + 79T^{2} \)
83 \( 1 + 1.41iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + 3iT - 97T^{2} \)
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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

−9.174628184612787133848603782142, −8.931700906418901317474974980039, −7.46878378496480525444242624032, −6.90670221465238658527995476753, −5.81392971546344827669767195182, −5.65111146848112257354826223966, −4.38204568320391849710925709148, −3.34312120071628030093462920203, −2.28117007132935914284038724616, −1.49420487921238821708445929026, 0.832724645568873429735610497587, 1.76002275331648304430222086535, 3.10262559583268932757745261390, 4.08921211569861187159800963719, 4.82280637084575706822080377461, 5.92298764961018639483372774224, 6.48763420572749446520304207625, 7.28377370986645533499435291240, 8.252977426187120605076059269027, 8.928760768047192673470852083052

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