Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $0.936 - 0.351i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 1.25i·5-s − 7-s − 1.55i·11-s + 1.07i·13-s − 0.0158·17-s − 2.35i·19-s + 5.95·23-s + 3.42·25-s + 0.469i·29-s − 1.69·31-s − 1.25i·35-s + 4.59i·37-s − 12.2·41-s + 1.97i·43-s + 7.12·47-s + ⋯
L(s)  = 1  + 0.560i·5-s − 0.377·7-s − 0.469i·11-s + 0.297i·13-s − 0.00383·17-s − 0.539i·19-s + 1.24·23-s + 0.685·25-s + 0.0872i·29-s − 0.303·31-s − 0.211i·35-s + 0.755i·37-s − 1.90·41-s + 0.301i·43-s + 1.03·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 - 0.351i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.936 - 0.351i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.936 - 0.351i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (3025, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 6048,\ (\ :1/2),\ 0.936 - 0.351i)\)
\(L(1)\)  \(\approx\)  \(1.790449434\)
\(L(\frac12)\)  \(\approx\)  \(1.790449434\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + T \)
good5 \( 1 - 1.25iT - 5T^{2} \)
11 \( 1 + 1.55iT - 11T^{2} \)
13 \( 1 - 1.07iT - 13T^{2} \)
17 \( 1 + 0.0158T + 17T^{2} \)
19 \( 1 + 2.35iT - 19T^{2} \)
23 \( 1 - 5.95T + 23T^{2} \)
29 \( 1 - 0.469iT - 29T^{2} \)
31 \( 1 + 1.69T + 31T^{2} \)
37 \( 1 - 4.59iT - 37T^{2} \)
41 \( 1 + 12.2T + 41T^{2} \)
43 \( 1 - 1.97iT - 43T^{2} \)
47 \( 1 - 7.12T + 47T^{2} \)
53 \( 1 - 1.86iT - 53T^{2} \)
59 \( 1 + 8.54iT - 59T^{2} \)
61 \( 1 - 3.92iT - 61T^{2} \)
67 \( 1 + 12.7iT - 67T^{2} \)
71 \( 1 + 4.22T + 71T^{2} \)
73 \( 1 - 6.51T + 73T^{2} \)
79 \( 1 - 6.15T + 79T^{2} \)
83 \( 1 + 8.88iT - 83T^{2} \)
89 \( 1 - 0.240T + 89T^{2} \)
97 \( 1 - 5.86T + 97T^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.146752727813428053980023784341, −7.20406124009224807671305230589, −6.77120546165381010463162308474, −6.13779024789013632878485232872, −5.19184092416025680091216796689, −4.57793883010464135109020104423, −3.39459370963556896072620136075, −3.05339493517896461807505471307, −1.97163022643808585003933443610, −0.73563157676183701658186242469, 0.67895225070283211855505521950, 1.72096438245081168256825850256, 2.77286209035130921862410793246, 3.59889176160564364116224336572, 4.43886767576019913542757609417, 5.21406401796810329640418909210, 5.75032021361072235865197404191, 6.80429273185866050292113580432, 7.18772987111700456360479338332, 8.118793354826855854052775418019

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