Properties

Degree 2
Conductor $ 2^{3} \cdot 3^{3} \cdot 7 $
Sign $0.249 + 0.968i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.40 + 0.118i)2-s + (1.97 − 0.335i)4-s + 3.46i·5-s + 7-s + (−2.73 + 0.707i)8-s + (−0.411 − 4.88i)10-s − 3.31i·11-s − 3.10i·13-s + (−1.40 + 0.118i)14-s + (3.77 − 1.32i)16-s + 1.40·17-s − 4.80i·19-s + (1.16 + 6.82i)20-s + (0.394 + 4.67i)22-s − 8.79·23-s + ⋯
L(s)  = 1  + (−0.996 + 0.0841i)2-s + (0.985 − 0.167i)4-s + 1.54i·5-s + 0.377·7-s + (−0.968 + 0.249i)8-s + (−0.130 − 1.54i)10-s − 1.00i·11-s − 0.862i·13-s + (−0.376 + 0.0317i)14-s + (0.943 − 0.330i)16-s + 0.340·17-s − 1.10i·19-s + (0.259 + 1.52i)20-s + (0.0841 + 0.997i)22-s − 1.83·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.249 + 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.249 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.249 + 0.968i$
motivic weight  =  \(1\)
character  :  $\chi_{1512} (757, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1512,\ (\ :1/2),\ 0.249 + 0.968i)\)
\(L(1)\)  \(\approx\)  \(0.6660891606\)
\(L(\frac12)\)  \(\approx\)  \(0.6660891606\)
\(L(\frac{3}{2})\)   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 \( 1 + (1.40 - 0.118i)T \)
3 \( 1 \)
7 \( 1 - T \)
good5 \( 1 - 3.46iT - 5T^{2} \)
11 \( 1 + 3.31iT - 11T^{2} \)
13 \( 1 + 3.10iT - 13T^{2} \)
17 \( 1 - 1.40T + 17T^{2} \)
19 \( 1 + 4.80iT - 19T^{2} \)
23 \( 1 + 8.79T + 23T^{2} \)
29 \( 1 + 9.87iT - 29T^{2} \)
31 \( 1 + 7.83T + 31T^{2} \)
37 \( 1 + 5.42iT - 37T^{2} \)
41 \( 1 - 11.5T + 41T^{2} \)
43 \( 1 + 7.03iT - 43T^{2} \)
47 \( 1 + 11.2T + 47T^{2} \)
53 \( 1 - 6.51iT - 53T^{2} \)
59 \( 1 + 3.89iT - 59T^{2} \)
61 \( 1 - 9.42iT - 61T^{2} \)
67 \( 1 - 0.909iT - 67T^{2} \)
71 \( 1 - 6.42T + 71T^{2} \)
73 \( 1 - 1.56T + 73T^{2} \)
79 \( 1 + 11.1T + 79T^{2} \)
83 \( 1 + 0.370iT - 83T^{2} \)
89 \( 1 + 4.85T + 89T^{2} \)
97 \( 1 - 1.63T + 97T^{2} \)
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

−9.409737294485442637552236172492, −8.364709346596747027171640183063, −7.71301584382030285228323784859, −7.13891510500186505496268987597, −6.05431443419561598601421626332, −5.73386407818846581483823609511, −3.88622261305751196215819124216, −2.91639927383210131644996732128, −2.15206964189892734735405585052, −0.35971477758597765316399104843, 1.41379310254297428079622939298, 1.92522915630097362714938372081, 3.69210532493531815849670572869, 4.64997617561910884144595544115, 5.55516586117261252156861213532, 6.52720684748974526556180881799, 7.61458574696717445647773101041, 8.113039080631577431200356652079, 8.891000091110648568699728813198, 9.571119231474322397831216473097

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