Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $-0.0430 - 0.999i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 3.11i·5-s − 7-s + 2.37i·11-s − 1.09i·13-s − 3.69·17-s + 1.08i·19-s + 4.87·23-s − 4.69·25-s − 1.59i·29-s − 7.45·31-s + 3.11i·35-s + 4.61i·37-s + 0.0380·41-s − 11.0i·43-s + 0.337·47-s + ⋯
L(s)  = 1  − 1.39i·5-s − 0.377·7-s + 0.715i·11-s − 0.302i·13-s − 0.895·17-s + 0.248i·19-s + 1.01·23-s − 0.939·25-s − 0.296i·29-s − 1.33·31-s + 0.526i·35-s + 0.758i·37-s + 0.00594·41-s − 1.68i·43-s + 0.0491·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0430 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0430 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $-0.0430 - 0.999i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (3025, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ -0.0430 - 0.999i)$
$L(1)$  $\approx$  $0.5792761787$
$L(\frac12)$  $\approx$  $0.5792761787$
$L(\frac{3}{2})$   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 \( 1 + 3.11iT - 5T^{2} \)
11 \( 1 - 2.37iT - 11T^{2} \)
13 \( 1 + 1.09iT - 13T^{2} \)
17 \( 1 + 3.69T + 17T^{2} \)
19 \( 1 - 1.08iT - 19T^{2} \)
23 \( 1 - 4.87T + 23T^{2} \)
29 \( 1 + 1.59iT - 29T^{2} \)
31 \( 1 + 7.45T + 31T^{2} \)
37 \( 1 - 4.61iT - 37T^{2} \)
41 \( 1 - 0.0380T + 41T^{2} \)
43 \( 1 + 11.0iT - 43T^{2} \)
47 \( 1 - 0.337T + 47T^{2} \)
53 \( 1 - 8.14iT - 53T^{2} \)
59 \( 1 - 15.0iT - 59T^{2} \)
61 \( 1 + 5.53iT - 61T^{2} \)
67 \( 1 - 7.70iT - 67T^{2} \)
71 \( 1 + 13.5T + 71T^{2} \)
73 \( 1 + 14.6T + 73T^{2} \)
79 \( 1 + 5.18T + 79T^{2} \)
83 \( 1 + 0.138iT - 83T^{2} \)
89 \( 1 - 11.9T + 89T^{2} \)
97 \( 1 + 5.30T + 97T^{2} \)
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

−8.480439057657471296502004613309, −7.43963088859165527775519029824, −7.02978637784989102082602740782, −5.95316886242379181564899385613, −5.37303212634277361069981633953, −4.59407336050026934505849399247, −4.09099345139468052189400363407, −2.99744421414545494525699732766, −1.95965453744634703098334421335, −1.04332246970928010380644819562, 0.15657060296259254985112648467, 1.73167557124131294243240433345, 2.77181065279785400713172339292, 3.24187860342056595227488426671, 4.08054445836580895786996332474, 5.05734120652270746401177607860, 5.96531020196926784838025492619, 6.56332383449769391361556537556, 7.07114499660954711069579413806, 7.70187449559542391778265195046

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