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Properties

Label 2-240-15.8-c1-0-2
Degree $2$
Conductor $240$
Sign $0.0618 - 0.998i$
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  + (0.292 + 1.70i)3-s + (0.707 + 2.12i)5-s + (1 − i)7-s + (−2.82 + i)9-s + 1.41i·11-s + (−3.41 + 1.82i)15-s + (−1.41 − 1.41i)17-s + 4i·19-s + (2 + 1.41i)21-s + (2.82 − 2.82i)23-s + (−3.99 + 3i)25-s + (−2.53 − 4.53i)27-s + 7.07·29-s + 2·31-s + (−2.41 + 0.414i)33-s + ⋯
L(s)  = 1  + (0.169 + 0.985i)3-s + (0.316 + 0.948i)5-s + (0.377 − 0.377i)7-s + (−0.942 + 0.333i)9-s + 0.426i·11-s + (−0.881 + 0.472i)15-s + (−0.342 − 0.342i)17-s + 0.917i·19-s + (0.436 + 0.308i)21-s + (0.589 − 0.589i)23-s + (−0.799 + 0.600i)25-s + (−0.487 − 0.872i)27-s + 1.31·29-s + 0.359·31-s + (−0.420 + 0.0721i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0618 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0618 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $0.0618 - 0.998i$
Analytic conductor: \(1.91640\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :1/2),\ 0.0618 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.978158 + 0.919405i\)
\(L(\frac12)\) \(\approx\) \(0.978158 + 0.919405i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.292 - 1.70i)T \)
5 \( 1 + (-0.707 - 2.12i)T \)
good7 \( 1 + (-1 + i)T - 7iT^{2} \)
11 \( 1 - 1.41iT - 11T^{2} \)
13 \( 1 + 13iT^{2} \)
17 \( 1 + (1.41 + 1.41i)T + 17iT^{2} \)
19 \( 1 - 4iT - 19T^{2} \)
23 \( 1 + (-2.82 + 2.82i)T - 23iT^{2} \)
29 \( 1 - 7.07T + 29T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 + (-6 + 6i)T - 37iT^{2} \)
41 \( 1 + 5.65iT - 41T^{2} \)
43 \( 1 + (6 + 6i)T + 43iT^{2} \)
47 \( 1 + 47iT^{2} \)
53 \( 1 + (-2.82 + 2.82i)T - 53iT^{2} \)
59 \( 1 - 9.89T + 59T^{2} \)
61 \( 1 + 6T + 61T^{2} \)
67 \( 1 + (-4 + 4i)T - 67iT^{2} \)
71 \( 1 + 14.1iT - 71T^{2} \)
73 \( 1 + (5 + 5i)T + 73iT^{2} \)
79 \( 1 - 6iT - 79T^{2} \)
83 \( 1 + (8.48 - 8.48i)T - 83iT^{2} \)
89 \( 1 + 2.82T + 89T^{2} \)
97 \( 1 + (-3 + 3i)T - 97iT^{2} \)
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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

−12.16447141339352205660376059511, −11.09182980429537591883014956127, −10.43511153928073453256784670117, −9.709039525516812508318820575832, −8.567044678880102270845229872779, −7.41233023072329660843655268632, −6.23713535916788691914735255083, −4.95608830693979098888685878477, −3.80313014105359759317226301696, −2.48219697651783341661105068535, 1.22294184313253671895092966857, 2.74865019178506398584206151122, 4.69688962225048418572061635101, 5.80466371356607943799907301478, 6.84125817143758795961452044454, 8.220307151502032302784688510121, 8.656557568382396809254784069579, 9.777768735286923549198064032291, 11.32549753105443260022881061788, 11.91500856088757303914227764973

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