↑

Properties

Label 2-3520-220.43-c0-0-1
Degree $2$
Conductor $3520$
Sign $-0.473 - 0.880i$
Analytic cond. $1.75670$
Root an. cond. $1.32540$
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  + (−1.36 + 1.36i)3-s + (0.866 + 0.5i)5-s − 2.73i·9-s − i·11-s + (−1.86 + 0.499i)15-s + (−0.366 + 0.366i)23-s + (0.499 + 0.866i)25-s + (2.36 + 2.36i)27-s + i·31-s + (1.36 + 1.36i)33-s + (−1.36 + 1.36i)37-s + (1.36 − 2.36i)45-s + (1 + i)47-s + i·49-s + (1 + i)53-s + ⋯
L(s)  = 1  + (−1.36 + 1.36i)3-s + (0.866 + 0.5i)5-s − 2.73i·9-s − i·11-s + (−1.86 + 0.499i)15-s + (−0.366 + 0.366i)23-s + (0.499 + 0.866i)25-s + (2.36 + 2.36i)27-s + i·31-s + (1.36 + 1.36i)33-s + (−1.36 + 1.36i)37-s + (1.36 − 2.36i)45-s + (1 + i)47-s + i·49-s + (1 + i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.473 - 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.473 - 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3520\)    =    \(2^{6} \cdot 5 \cdot 11\)
Sign: $-0.473 - 0.880i$
Analytic conductor: \(1.75670\)
Root analytic conductor: \(1.32540\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3520} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3520,\ (\ :0),\ -0.473 - 0.880i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8238260165\)
\(L(\frac12)\) \(\approx\) \(0.8238260165\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-0.866 - 0.5i)T \)
11 \( 1 + iT \)
good3 \( 1 + (1.36 - 1.36i)T - iT^{2} \)
7 \( 1 - iT^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (0.366 - 0.366i)T - iT^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - iT - T^{2} \)
37 \( 1 + (1.36 - 1.36i)T - iT^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (-1 - i)T + iT^{2} \)
53 \( 1 + (-1 - i)T + iT^{2} \)
59 \( 1 - T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + (0.366 + 0.366i)T + iT^{2} \)
71 \( 1 - 1.73iT - T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + iT - T^{2} \)
97 \( 1 + (-0.366 + 0.366i)T - iT^{2} \)
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

−9.125276276354412162330659714198, −8.635227140235580539271911087453, −7.21372299239001030401981284688, −6.44078641957222857156920400055, −5.84667250934985024488891245195, −5.37619916064942311692399323869, −4.56097601816196122610776822627, −3.62448002906489498602673664194, −2.90167105056054528659212652085, −1.20395041384012170671144681015, 0.64287471204707743279962651127, 1.91819321434550660751914295908, 2.21895088532058530696036994157, 4.12506895786280931140142993512, 5.15651593624672119896194799625, 5.47505734883305686562763179559, 6.32056243753819416547568577285, 6.90946353560199157889544809793, 7.51645910372355878745417187935, 8.354966574490130060218613218875

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