Properties

Label 2-240-12.11-c1-0-4
Degree $2$
Conductor $240$
Sign $0.866 - 0.5i$
Analytic cond. $1.91640$
Root an. cond. $1.38434$
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  + 1.73·3-s + i·5-s + 3.46i·7-s + 2.99·9-s − 3.46·11-s + 4·13-s + 1.73i·15-s − 6i·17-s − 3.46i·19-s + 5.99i·21-s − 3.46·23-s − 25-s + 5.19·27-s − 6i·29-s + 3.46i·31-s + ⋯
L(s)  = 1  + 1.00·3-s + 0.447i·5-s + 1.30i·7-s + 0.999·9-s − 1.04·11-s + 1.10·13-s + 0.447i·15-s − 1.45i·17-s − 0.794i·19-s + 1.30i·21-s − 0.722·23-s − 0.200·25-s + 1.00·27-s − 1.11i·29-s + 0.622i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $0.866 - 0.5i$
Analytic conductor: \(1.91640\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :1/2),\ 0.866 - 0.5i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.60285 + 0.429484i\)
\(L(\frac12)\) \(\approx\) \(1.60285 + 0.429484i\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.73T \)
5 \( 1 - iT \)
good7 \( 1 - 3.46iT - 7T^{2} \)
11 \( 1 + 3.46T + 11T^{2} \)
13 \( 1 - 4T + 13T^{2} \)
17 \( 1 + 6iT - 17T^{2} \)
19 \( 1 + 3.46iT - 19T^{2} \)
23 \( 1 + 3.46T + 23T^{2} \)
29 \( 1 + 6iT - 29T^{2} \)
31 \( 1 - 3.46iT - 31T^{2} \)
37 \( 1 + 4T + 37T^{2} \)
41 \( 1 - 12iT - 41T^{2} \)
43 \( 1 + 6.92iT - 43T^{2} \)
47 \( 1 - 3.46T + 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 + 3.46T + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 + 6.92iT - 67T^{2} \)
71 \( 1 + 13.8T + 71T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 - 10.3iT - 79T^{2} \)
83 \( 1 - 10.3T + 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 - 10T + 97T^{2} \)
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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

−12.24952197253693054413183446311, −11.28440144404897896721013368437, −10.13044822963774145244961567715, −9.168959417317104462128556340560, −8.396308964178784381032769143786, −7.42638601916103125654202861701, −6.15389010861668252781664991146, −4.86576934210970850240870448791, −3.19256541621760682779986301493, −2.31495790927560255390711593253, 1.59071496736681189463183431482, 3.50070174025271936712968003497, 4.31480145888520347780293162814, 5.95949156396067175979958237317, 7.39696918254458503511119639952, 8.105698884053556510425886759223, 8.961561817606874687260306539237, 10.40865450868908857605531772836, 10.56769882685487011881329995865, 12.36655202303363276666116213589

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