↑

Properties

Label 2-3420-285.227-c0-0-3
Degree $2$
Conductor $3420$
Sign $0.948 + 0.318i$
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  + (−0.965 − 0.258i)5-s + (0.366 + 0.366i)7-s − 0.517i·11-s + (0.707 − 0.707i)17-s + i·19-s + (0.866 + 0.499i)25-s + (−0.258 − 0.448i)35-s + (0.366 − 0.366i)43-s + (1.22 − 1.22i)47-s − 0.732i·49-s + (−0.133 + 0.499i)55-s + 61-s + (1.36 − 1.36i)73-s + (0.189 − 0.189i)77-s + (1.41 + 1.41i)83-s + ⋯
L(s)  = 1  + (−0.965 − 0.258i)5-s + (0.366 + 0.366i)7-s − 0.517i·11-s + (0.707 − 0.707i)17-s + i·19-s + (0.866 + 0.499i)25-s + (−0.258 − 0.448i)35-s + (0.366 − 0.366i)43-s + (1.22 − 1.22i)47-s − 0.732i·49-s + (−0.133 + 0.499i)55-s + 61-s + (1.36 − 1.36i)73-s + (0.189 − 0.189i)77-s + (1.41 + 1.41i)83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 + 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 + 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3420\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $0.948 + 0.318i$
Analytic conductor: \(1.70680\)
Root analytic conductor: \(1.30644\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3420} (3077, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3420,\ (\ :0),\ 0.948 + 0.318i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.106903756\)
\(L(\frac12)\) \(\approx\) \(1.106903756\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (0.965 + 0.258i)T \)
19 \( 1 - iT \)
good7 \( 1 + (-0.366 - 0.366i)T + iT^{2} \)
11 \( 1 + 0.517iT - T^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (-0.366 + 0.366i)T - iT^{2} \)
47 \( 1 + (-1.22 + 1.22i)T - iT^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T + T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-1.36 + 1.36i)T - iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + (-1.41 - 1.41i)T + iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - iT^{2} \)
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

−8.619153505256773511881147184385, −8.015761479981538766662856833083, −7.44647316888822063009452726658, −6.57153030650901463822531229541, −5.54056366887859392811731700540, −5.04755445460646453526180093213, −3.95333612918543419515526058365, −3.39082753813057231526240897878, −2.23306346387909462543683826202, −0.872574791903430145345897614743, 1.03259088009734866975337964001, 2.40441943382496337813622217516, 3.38880843961196571299635815475, 4.22087013652295871497616924232, 4.80487260198088668693569958219, 5.83846848703630818465684277273, 6.77030659688749183217083252340, 7.42918818859619762269225536912, 7.945997613969061483803753472174, 8.703550360843151522931628778995

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