Properties

Label 2-240-12.11-c1-0-1
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

\(L(1)\) \(\approx\) \(0.952230 + 0.255149i\)
\(L(\frac12)\) \(\approx\) \(0.952230 + 0.255149i\)
\(L(\frac{3}{2})\) 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

−12.30578051138520551680547470918, −11.30921205938978454033863562212, −10.54791302366918601861539270808, −9.078145881665742489829722192453, −8.639572087645957742061193668300, −6.89821712016416042846394111613, −6.00233065385101713047465352708, −5.15382809774023793858006631965, −3.77119613111660768776247420645, −1.56201704167621556304373127495, 1.10465757482853634816726199647, 3.61720455341027874233140882971, 4.63300665654680173785481794124, 6.14226000498991735911090332878, 6.84903482209633190987410977549, 7.82236899021482210033057312024, 9.466257857699990188138416056947, 10.26027355890494776127937633554, 11.29545099402316389955785477197, 11.63455029835239213040350207982

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