Properties

Label 2-2520-2520.797-c0-0-4
Degree $2$
Conductor $2520$
Sign $-0.944 + 0.329i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.965 − 0.258i)2-s + (−0.923 + 0.382i)3-s + (0.866 + 0.499i)4-s + (−0.793 − 0.608i)5-s + (0.991 − 0.130i)6-s + (−0.258 + 0.965i)7-s + (−0.707 − 0.707i)8-s + (0.707 − 0.707i)9-s + (0.608 + 0.793i)10-s + (−0.991 − 0.130i)12-s + (−0.198 − 0.739i)13-s + (0.499 − 0.866i)14-s + (0.965 + 0.258i)15-s + (0.500 + 0.866i)16-s + (−0.866 + 0.500i)18-s + 0.261i·19-s + ⋯
L(s)  = 1  + (−0.965 − 0.258i)2-s + (−0.923 + 0.382i)3-s + (0.866 + 0.499i)4-s + (−0.793 − 0.608i)5-s + (0.991 − 0.130i)6-s + (−0.258 + 0.965i)7-s + (−0.707 − 0.707i)8-s + (0.707 − 0.707i)9-s + (0.608 + 0.793i)10-s + (−0.991 − 0.130i)12-s + (−0.198 − 0.739i)13-s + (0.499 − 0.866i)14-s + (0.965 + 0.258i)15-s + (0.500 + 0.866i)16-s + (−0.866 + 0.500i)18-s + 0.261i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 + 0.329i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 + 0.329i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2520\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.944 + 0.329i$
Analytic conductor: \(1.25764\)
Root analytic conductor: \(1.12144\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2520} (797, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2520,\ (\ :0),\ -0.944 + 0.329i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.06116482391\)
\(L(\frac12)\) \(\approx\) \(0.06116482391\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 + 0.258i)T \)
3 \( 1 + (0.923 - 0.382i)T \)
5 \( 1 + (0.793 + 0.608i)T \)
7 \( 1 + (0.258 - 0.965i)T \)
good11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.198 + 0.739i)T + (-0.866 + 0.5i)T^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 - 0.261iT - T^{2} \)
23 \( 1 + (0.5 - 0.133i)T + (0.866 - 0.5i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.866 - 0.5i)T^{2} \)
47 \( 1 + (-0.866 - 0.5i)T^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + (0.923 - 1.60i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.991 + 1.71i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.866 + 0.5i)T^{2} \)
71 \( 1 + 1.93iT - T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 + (1.67 - 0.965i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.478 + 1.78i)T + (-0.866 - 0.5i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.866 + 0.5i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.923330469277928694030254265358, −8.054341696052876359262324783332, −7.45448119827675038374746222012, −6.41320310074183509294204656329, −5.74271532882434546586977065826, −4.88321178369971366222336415595, −3.82637418140102510407590464804, −2.93616050639384359785855766337, −1.50530196165526621226038457796, −0.06844353340600499167192026095, 1.30760216067407041436855636006, 2.59714772198529918828299973845, 3.89336326229843794834948700530, 4.76528988267683319845523223389, 5.94672292971556185354070205352, 6.66352397101990503965825346727, 7.14796567317980102112324184251, 7.69552221554201742962573844773, 8.483166167798445163332667330685, 9.604214196532454731348261029500

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