Properties

Label 2-4032-12.11-c1-0-29
Degree $2$
Conductor $4032$
Sign $0.816 - 0.577i$
Analytic cond. $32.1956$
Root an. cond. $5.67412$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3.31i·5-s i·7-s + 4.72·11-s + 4.97·13-s − 0.484i·17-s + 2.29i·19-s + 7.97·23-s − 5.97·25-s + 1.41i·29-s − 7.66i·31-s + 3.31·35-s + 2.39·37-s − 6.55i·41-s − 5.37i·43-s + 6.21·47-s + ⋯
L(s)  = 1  + 1.48i·5-s − 0.377i·7-s + 1.42·11-s + 1.38·13-s − 0.117i·17-s + 0.525i·19-s + 1.66·23-s − 1.19·25-s + 0.262i·29-s − 1.37i·31-s + 0.560·35-s + 0.393·37-s − 1.02i·41-s − 0.819i·43-s + 0.906·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
Sign: $0.816 - 0.577i$
Analytic conductor: \(32.1956\)
Root analytic conductor: \(5.67412\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4032} (575, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4032,\ (\ :1/2),\ 0.816 - 0.577i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.445467761\)
\(L(\frac12)\) \(\approx\) \(2.445467761\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + iT \)
good5 \( 1 - 3.31iT - 5T^{2} \)
11 \( 1 - 4.72T + 11T^{2} \)
13 \( 1 - 4.97T + 13T^{2} \)
17 \( 1 + 0.484iT - 17T^{2} \)
19 \( 1 - 2.29iT - 19T^{2} \)
23 \( 1 - 7.97T + 23T^{2} \)
29 \( 1 - 1.41iT - 29T^{2} \)
31 \( 1 + 7.66iT - 31T^{2} \)
37 \( 1 - 2.39T + 37T^{2} \)
41 \( 1 + 6.55iT - 41T^{2} \)
43 \( 1 + 5.37iT - 43T^{2} \)
47 \( 1 - 6.21T + 47T^{2} \)
53 \( 1 + 1.00iT - 53T^{2} \)
59 \( 1 + 1.38T + 59T^{2} \)
61 \( 1 + 13.6T + 61T^{2} \)
67 \( 1 + 3.27iT - 67T^{2} \)
71 \( 1 - 3.34T + 71T^{2} \)
73 \( 1 + 2.10T + 73T^{2} \)
79 \( 1 + 12.0iT - 79T^{2} \)
83 \( 1 - 3.24T + 83T^{2} \)
89 \( 1 + 5.72iT - 89T^{2} \)
97 \( 1 - 12.0T + 97T^{2} \)
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

−8.646363666160484527382602038718, −7.46223067263895439521490789935, −7.13298403690974437014487588763, −6.23297268385253889685720221186, −5.97129205032038572526985117408, −4.57111570898596251303621258598, −3.64355450074110639381330563881, −3.31600237701645880451615213888, −2.09748041020420391754801955811, −0.984519622445090559170399204414, 1.04866839736139433784435951726, 1.41086467593643744262265600826, 2.94859200500914392048786501511, 3.90024403737894514009159396125, 4.63159417791880573544708459129, 5.28766893528628699347634635323, 6.20049234615255610906225957070, 6.72377289320838529218248628900, 7.83564390756393400708377530215, 8.616133793787232728291373940565

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