↑

Properties

Label 2-3660-3660.1499-c0-0-1
Degree $2$
Conductor $3660$
Sign $-0.0217 - 0.999i$
Analytic cond. $1.82657$
Root an. cond. $1.35150$
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  + (−0.544 + 0.838i)2-s + (−0.933 − 0.358i)3-s + (−0.406 − 0.913i)4-s + (0.104 − 0.994i)5-s + (0.809 − 0.587i)6-s + (−1.61 + 0.0846i)7-s + (0.987 + 0.156i)8-s + (0.743 + 0.669i)9-s + (0.777 + 0.629i)10-s + (0.0523 + 0.998i)12-s + (0.809 − 1.40i)14-s + (−0.453 + 0.891i)15-s + (−0.669 + 0.743i)16-s + (−0.965 + 0.258i)18-s + (−0.951 + 0.309i)20-s + (1.53 + 0.5i)21-s + ⋯
L(s)  = 1  + (−0.544 + 0.838i)2-s + (−0.933 − 0.358i)3-s + (−0.406 − 0.913i)4-s + (0.104 − 0.994i)5-s + (0.809 − 0.587i)6-s + (−1.61 + 0.0846i)7-s + (0.987 + 0.156i)8-s + (0.743 + 0.669i)9-s + (0.777 + 0.629i)10-s + (0.0523 + 0.998i)12-s + (0.809 − 1.40i)14-s + (−0.453 + 0.891i)15-s + (−0.669 + 0.743i)16-s + (−0.965 + 0.258i)18-s + (−0.951 + 0.309i)20-s + (1.53 + 0.5i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3660 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0217 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3660 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0217 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3660\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 61\)
Sign: $-0.0217 - 0.999i$
Analytic conductor: \(1.82657\)
Root analytic conductor: \(1.35150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3660} (1499, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3660,\ (\ :0),\ -0.0217 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2875148832\)
\(L(\frac12)\) \(\approx\) \(0.2875148832\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.544 - 0.838i)T \)
3 \( 1 + (0.933 + 0.358i)T \)
5 \( 1 + (-0.104 + 0.994i)T \)
61 \( 1 + (0.951 + 0.309i)T \)
good7 \( 1 + (1.61 - 0.0846i)T + (0.994 - 0.104i)T^{2} \)
11 \( 1 - iT^{2} \)
13 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (-0.743 - 0.669i)T^{2} \)
19 \( 1 + (-0.104 + 0.994i)T^{2} \)
23 \( 1 + (0.410 - 0.0650i)T + (0.951 - 0.309i)T^{2} \)
29 \( 1 + (1.80 - 0.483i)T + (0.866 - 0.5i)T^{2} \)
31 \( 1 + (-0.406 + 0.913i)T^{2} \)
37 \( 1 + (-0.587 - 0.809i)T^{2} \)
41 \( 1 + (-0.413 - 1.27i)T + (-0.809 + 0.587i)T^{2} \)
43 \( 1 + (-1.70 + 0.654i)T + (0.743 - 0.669i)T^{2} \)
47 \( 1 + (0.786 + 0.453i)T + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.951 + 0.309i)T^{2} \)
59 \( 1 + (-0.406 - 0.913i)T^{2} \)
67 \( 1 + (-0.629 - 0.777i)T + (-0.207 + 0.978i)T^{2} \)
71 \( 1 + (0.207 + 0.978i)T^{2} \)
73 \( 1 + (0.978 - 0.207i)T^{2} \)
79 \( 1 + (0.743 - 0.669i)T^{2} \)
83 \( 1 + (-0.370 - 1.74i)T + (-0.913 + 0.406i)T^{2} \)
89 \( 1 + (1.72 - 0.877i)T + (0.587 - 0.809i)T^{2} \)
97 \( 1 + (0.913 + 0.406i)T^{2} \)
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

−8.994136914088950025444059490751, −8.030375259589081105906697610239, −7.33279595689817276694855683049, −6.63787900804918441120094613056, −5.90469290090326851045509344622, −5.55903656694738824042851776509, −4.61528304773439207557584888996, −3.73393365512163249943745639801, −2.08434931510247036810405839483, −0.889359569763891282004588105620, 0.29773216125136410803983586969, 1.98937995724963939258534099646, 3.08143811049448918178496193654, 3.66374712451128184335289627875, 4.40835023874524695356802433182, 5.79516029893186753736335197930, 6.20891710194486346225769047921, 7.19193775467149698357840507635, 7.52992095510564901073957788079, 8.941154136704404962088077424633

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