↑

Properties

Label 2-3420-3420.1139-c0-0-13
Degree $2$
Conductor $3420$
Sign $-0.766 + 0.642i$
Analytic cond. $1.70680$
Root an. cond. $1.30644$
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  + (−0.793 + 0.608i)2-s + (−0.991 + 0.130i)3-s + (0.258 − 0.965i)4-s + (−0.866 − 0.5i)5-s + (0.707 − 0.707i)6-s + (0.382 + 0.923i)8-s + (0.965 − 0.258i)9-s + (0.991 − 0.130i)10-s + (−0.965 − 1.67i)11-s + (−0.130 + 0.991i)12-s + (0.793 − 1.37i)13-s + (0.923 + 0.382i)15-s + (−0.866 − 0.499i)16-s + (−0.608 + 0.793i)18-s − i·19-s + (−0.707 + 0.707i)20-s + ⋯
L(s)  = 1  + (−0.793 + 0.608i)2-s + (−0.991 + 0.130i)3-s + (0.258 − 0.965i)4-s + (−0.866 − 0.5i)5-s + (0.707 − 0.707i)6-s + (0.382 + 0.923i)8-s + (0.965 − 0.258i)9-s + (0.991 − 0.130i)10-s + (−0.965 − 1.67i)11-s + (−0.130 + 0.991i)12-s + (0.793 − 1.37i)13-s + (0.923 + 0.382i)15-s + (−0.866 − 0.499i)16-s + (−0.608 + 0.793i)18-s − i·19-s + (−0.707 + 0.707i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3420\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $-0.766 + 0.642i$
Analytic conductor: \(1.70680\)
Root analytic conductor: \(1.30644\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3420} (1139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3420,\ (\ :0),\ -0.766 + 0.642i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2778674555\)
\(L(\frac12)\) \(\approx\) \(0.2778674555\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.793 - 0.608i)T \)
3 \( 1 + (0.991 - 0.130i)T \)
5 \( 1 + (0.866 + 0.5i)T \)
19 \( 1 + iT \)
good7 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.965 + 1.67i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.793 + 1.37i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + 1.21T + T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 - 0.261iT - T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.662 - 0.382i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (0.382 + 0.662i)T + (-0.5 + 0.866i)T^{2} \)
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

−8.499672702348182842522071093849, −7.81473625477386424153893556133, −7.10099126996055210612570563204, −6.21846217585160613215504647274, −5.36680853060462528779708702801, −5.23608015872748601556895091513, −3.93567619562219044686577470882, −2.91098441531836833994926093563, −1.07670041827541584109693642549, −0.30173604604116445074060521636, 1.51381337377558041525251840654, 2.33128486571312079422176039535, 3.71916953082679698350632634045, 4.26722058237406130779048461667, 5.14232054362049058849803506478, 6.48719405925634064022042612747, 6.91126111869581946863228076440, 7.62937222376212740829874532236, 8.165494249837084457191026898838, 9.240973503836242865995395033868

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