↑

Properties

Label 2-3192-3192.1637-c0-0-7
Degree $2$
Conductor $3192$
Sign $0.939 - 0.342i$
Analytic cond. $1.59301$
Root an. cond. $1.26214$
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  + (0.766 + 0.642i)2-s + (0.642 − 0.766i)3-s + (0.173 + 0.984i)4-s + (1.50 + 0.266i)5-s + (0.984 − 0.173i)6-s + (−0.5 − 0.866i)7-s + (−0.500 + 0.866i)8-s + (−0.173 − 0.984i)9-s + (0.984 + 1.17i)10-s + (0.866 + 0.500i)12-s + (−0.118 + 0.326i)13-s + (0.173 − 0.984i)14-s + (1.17 − 0.984i)15-s + (−0.939 + 0.342i)16-s + (0.500 − 0.866i)18-s + (−0.642 + 0.766i)19-s + ⋯
L(s)  = 1  + (0.766 + 0.642i)2-s + (0.642 − 0.766i)3-s + (0.173 + 0.984i)4-s + (1.50 + 0.266i)5-s + (0.984 − 0.173i)6-s + (−0.5 − 0.866i)7-s + (−0.500 + 0.866i)8-s + (−0.173 − 0.984i)9-s + (0.984 + 1.17i)10-s + (0.866 + 0.500i)12-s + (−0.118 + 0.326i)13-s + (0.173 − 0.984i)14-s + (1.17 − 0.984i)15-s + (−0.939 + 0.342i)16-s + (0.500 − 0.866i)18-s + (−0.642 + 0.766i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3192\)    =    \(2^{3} \cdot 3 \cdot 7 \cdot 19\)
Sign: $0.939 - 0.342i$
Analytic conductor: \(1.59301\)
Root analytic conductor: \(1.26214\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3192} (1637, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3192,\ (\ :0),\ 0.939 - 0.342i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.811785622\)
\(L(\frac12)\) \(\approx\) \(2.811785622\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.766 - 0.642i)T \)
3 \( 1 + (-0.642 + 0.766i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (0.642 - 0.766i)T \)
good5 \( 1 + (-1.50 - 0.266i)T + (0.939 + 0.342i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.118 - 0.326i)T + (-0.766 - 0.642i)T^{2} \)
17 \( 1 + (0.173 + 0.984i)T^{2} \)
23 \( 1 + (-1.26 + 0.223i)T + (0.939 - 0.342i)T^{2} \)
29 \( 1 + (-0.173 + 0.984i)T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (-0.766 + 0.642i)T^{2} \)
43 \( 1 + (0.939 + 0.342i)T^{2} \)
47 \( 1 + (0.173 - 0.984i)T^{2} \)
53 \( 1 + (0.939 - 0.342i)T^{2} \)
59 \( 1 + (0.524 + 0.439i)T + (0.173 + 0.984i)T^{2} \)
61 \( 1 + (-0.118 - 0.673i)T + (-0.939 + 0.342i)T^{2} \)
67 \( 1 + (0.173 - 0.984i)T^{2} \)
71 \( 1 + (-0.326 + 1.85i)T + (-0.939 - 0.342i)T^{2} \)
73 \( 1 + (-0.766 + 0.642i)T^{2} \)
79 \( 1 + (0.592 + 1.62i)T + (-0.766 + 0.642i)T^{2} \)
83 \( 1 + (1.62 - 0.939i)T + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (-0.766 - 0.642i)T^{2} \)
97 \( 1 + (0.173 + 0.984i)T^{2} \)
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

−8.859295475605796413847640324422, −7.900489967927115779190354845658, −7.18780932890153139609785933188, −6.51828935494866110683378979845, −6.19241149994916940387049266903, −5.24841271232930080551240095211, −4.18992108939880932943756093271, −3.26099658242100543389622156632, −2.51309619547779340185240810182, −1.54837548671595898967083528712, 1.60589006947095946729717713245, 2.60127565340429153397172884975, 2.90478187186551185716107912363, 4.11187673109320410880182398467, 5.09402192467690183025455248771, 5.44042058576542142959957938599, 6.23484028392094730539457824775, 7.06335125046196356972403798844, 8.564519644605829795806595742743, 9.025573795553814602669429538789

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