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Properties

Label 2-240-20.3-c1-0-2
Degree $2$
Conductor $240$
Sign $0.923 + 0.382i$
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  + (0.707 − 0.707i)3-s + (2.22 + 0.224i)5-s + (−0.317 − 0.317i)7-s − 1.00i·9-s − 1.41i·11-s + (2.44 + 2.44i)13-s + (1.73 − 1.41i)15-s + (−0.449 + 0.449i)17-s − 6.29·19-s − 0.449·21-s + (4.87 − 4.87i)23-s + (4.89 + i)25-s + (−0.707 − 0.707i)27-s − 5.34i·29-s + 8.48i·31-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (0.994 + 0.100i)5-s + (−0.120 − 0.120i)7-s − 0.333i·9-s − 0.426i·11-s + (0.679 + 0.679i)13-s + (0.447 − 0.365i)15-s + (−0.109 + 0.109i)17-s − 1.44·19-s − 0.0980·21-s + (1.01 − 1.01i)23-s + (0.979 + 0.200i)25-s + (−0.136 − 0.136i)27-s − 0.993i·29-s + 1.52i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $0.923 + 0.382i$
Analytic conductor: \(1.91640\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :1/2),\ 0.923 + 0.382i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.55329 - 0.308770i\)
\(L(\frac12)\) \(\approx\) \(1.55329 - 0.308770i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 + (-2.22 - 0.224i)T \)
good7 \( 1 + (0.317 + 0.317i)T + 7iT^{2} \)
11 \( 1 + 1.41iT - 11T^{2} \)
13 \( 1 + (-2.44 - 2.44i)T + 13iT^{2} \)
17 \( 1 + (0.449 - 0.449i)T - 17iT^{2} \)
19 \( 1 + 6.29T + 19T^{2} \)
23 \( 1 + (-4.87 + 4.87i)T - 23iT^{2} \)
29 \( 1 + 5.34iT - 29T^{2} \)
31 \( 1 - 8.48iT - 31T^{2} \)
37 \( 1 + (6.44 - 6.44i)T - 37iT^{2} \)
41 \( 1 + 10.8T + 41T^{2} \)
43 \( 1 + (6.29 - 6.29i)T - 43iT^{2} \)
47 \( 1 + (-8.34 - 8.34i)T + 47iT^{2} \)
53 \( 1 + (2 + 2i)T + 53iT^{2} \)
59 \( 1 + 9.89T + 59T^{2} \)
61 \( 1 + 6.89T + 61T^{2} \)
67 \( 1 + (0.635 + 0.635i)T + 67iT^{2} \)
71 \( 1 + 1.27iT - 71T^{2} \)
73 \( 1 + (-3 - 3i)T + 73iT^{2} \)
79 \( 1 - 4.09T + 79T^{2} \)
83 \( 1 + (-1.41 + 1.41i)T - 83iT^{2} \)
89 \( 1 + 2iT - 89T^{2} \)
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

−12.25085301898553382857709353532, −10.96675372922261122587733485618, −10.19627127630683128986039383806, −8.957441779999834581821019541305, −8.419252836866365376919238392687, −6.75714426837349592844789504020, −6.28573724775002233172618692228, −4.74136163389259876954214629147, −3.12650384198720997292700840579, −1.70356475293786891742151314392, 1.98657288960737003802095389447, 3.45816334435872300551510926922, 4.95092513728393703453515116454, 5.96655715716247665430064049251, 7.17622741399842327486074158044, 8.584161200933331682766112489417, 9.222435253148990891147578725666, 10.28705137145633960664213687126, 10.91455153977739358348825862026, 12.39119968332244650134618271111

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