Properties

Label 2-240-3.2-c2-0-14
Degree $2$
Conductor $240$
Sign $-0.995 + 0.0972i$
Analytic cond. $6.53952$
Root an. cond. $2.55724$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.291 − 2.98i)3-s + 2.23i·5-s − 4.46·7-s + (−8.82 + 1.74i)9-s − 17.8i·11-s − 11.0·13-s + (6.67 − 0.652i)15-s − 0.794i·17-s − 26.5·19-s + (1.30 + 13.3i)21-s + 14.9i·23-s − 5.00·25-s + (7.77 + 25.8i)27-s − 5.58i·29-s − 53.1·31-s + ⋯
L(s)  = 1  + (−0.0972 − 0.995i)3-s + 0.447i·5-s − 0.637·7-s + (−0.981 + 0.193i)9-s − 1.62i·11-s − 0.846·13-s + (0.445 − 0.0434i)15-s − 0.0467i·17-s − 1.39·19-s + (0.0619 + 0.634i)21-s + 0.649i·23-s − 0.200·25-s + (0.287 + 0.957i)27-s − 0.192i·29-s − 1.71·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0972i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.995 + 0.0972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $-0.995 + 0.0972i$
Analytic conductor: \(6.53952\)
Root analytic conductor: \(2.55724\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :1),\ -0.995 + 0.0972i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0269264 - 0.552710i\)
\(L(\frac12)\) \(\approx\) \(0.0269264 - 0.552710i\)
\(L(2)\) 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 \( 1 + (0.291 + 2.98i)T \)
5 \( 1 - 2.23iT \)
good7 \( 1 + 4.46T + 49T^{2} \)
11 \( 1 + 17.8iT - 121T^{2} \)
13 \( 1 + 11.0T + 169T^{2} \)
17 \( 1 + 0.794iT - 289T^{2} \)
19 \( 1 + 26.5T + 361T^{2} \)
23 \( 1 - 14.9iT - 529T^{2} \)
29 \( 1 + 5.58iT - 841T^{2} \)
31 \( 1 + 53.1T + 961T^{2} \)
37 \( 1 - 51.7T + 1.36e3T^{2} \)
41 \( 1 + 67.8iT - 1.68e3T^{2} \)
43 \( 1 - 40.8T + 1.84e3T^{2} \)
47 \( 1 - 12.3iT - 2.20e3T^{2} \)
53 \( 1 + 37.0iT - 2.80e3T^{2} \)
59 \( 1 + 61.0iT - 3.48e3T^{2} \)
61 \( 1 - 97.8T + 3.72e3T^{2} \)
67 \( 1 + 3.02T + 4.48e3T^{2} \)
71 \( 1 + 57.0iT - 5.04e3T^{2} \)
73 \( 1 + 31.4T + 5.32e3T^{2} \)
79 \( 1 - 2.16T + 6.24e3T^{2} \)
83 \( 1 + 13.0iT - 6.88e3T^{2} \)
89 \( 1 - 173. iT - 7.92e3T^{2} \)
97 \( 1 + 91.6T + 9.40e3T^{2} \)
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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

−11.39075325082377708283701565322, −10.77392520695997717074236115388, −9.387105534807122877386187712043, −8.365224067902590948018423769753, −7.34550821082630863228004763901, −6.37695927773937211626463361579, −5.57190243539116334808782534846, −3.56854509800781785202876637558, −2.32961368742133351827071929985, −0.27550598461632500629867947474, 2.44157449368364709390983224817, 4.12241848697964501109752299065, 4.85295323019559127988045927135, 6.17082556288374881022738946195, 7.40927845716401331914373397469, 8.752146324726368353010223935441, 9.638964366881985319707789066146, 10.19824048693222751081604702161, 11.32161523142629311773020032023, 12.52356497443791040135653426717

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