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Properties

Label 2-2520-840.299-c0-0-6
Degree $2$
Conductor $2520$
Sign $-0.292 + 0.956i$
Analytic cond. $1.25764$
Root an. cond. $1.12144$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + (−0.5 + 0.866i)5-s + (0.707 − 0.707i)7-s − 0.999i·8-s + (0.866 − 0.499i)10-s + (0.448 − 0.258i)11-s − 1.93i·13-s + (−0.965 + 0.258i)14-s + (−0.5 + 0.866i)16-s + (−0.866 − 0.5i)19-s − 0.999·20-s − 0.517·22-s + (−1.5 − 0.866i)23-s + (−0.499 − 0.866i)25-s + (−0.965 + 1.67i)26-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + (−0.5 + 0.866i)5-s + (0.707 − 0.707i)7-s − 0.999i·8-s + (0.866 − 0.499i)10-s + (0.448 − 0.258i)11-s − 1.93i·13-s + (−0.965 + 0.258i)14-s + (−0.5 + 0.866i)16-s + (−0.866 − 0.5i)19-s − 0.999·20-s − 0.517·22-s + (−1.5 − 0.866i)23-s + (−0.499 − 0.866i)25-s + (−0.965 + 1.67i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.292 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.292 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2520\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.292 + 0.956i$
Analytic conductor: \(1.25764\)
Root analytic conductor: \(1.12144\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2520} (1979, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2520,\ (\ :0),\ -0.292 + 0.956i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6142357908\)
\(L(\frac12)\) \(\approx\) \(0.6142357908\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-0.707 + 0.707i)T \)
good11 \( 1 + (-0.448 + 0.258i)T + (0.5 - 0.866i)T^{2} \)
13 \( 1 + 1.93iT - T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.965 - 1.67i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 - 0.517T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + T^{2} \)
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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

−8.620122807928384260136288486336, −8.149705917205951800720585116256, −7.61468838232533713679516408383, −6.78540233262014006190101645260, −6.05945313882672549321221121267, −4.66665754908628450691000349812, −3.75742697709939887861895654743, −3.04963537027478760152322040675, −2.00775101375048727644421955613, −0.53703348900339963145703615923, 1.58154652192447398187998656655, 2.07566503720676041327356890164, 4.01708366109892423749793215733, 4.58971482727373633547565068252, 5.63711456712415529308045863432, 6.24780050485837676856553394770, 7.27831442044283802170128861062, 7.87747801511242124519473351291, 8.666614790443279086323105697239, 9.149022989276011331648470699777

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