Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $0.894 + 0.447i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  i·2-s − 4-s + i·7-s + i·8-s + 4·11-s − 3i·13-s + 14-s + 16-s + 7i·17-s + 6·19-s − 4i·22-s − 9i·23-s − 3·26-s i·28-s − 3·29-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + 0.377i·7-s + 0.353i·8-s + 1.20·11-s − 0.832i·13-s + 0.267·14-s + 0.250·16-s + 1.69i·17-s + 1.37·19-s − 0.852i·22-s − 1.87i·23-s − 0.588·26-s − 0.188i·28-s − 0.557·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.894 + 0.447i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (2899, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ 0.894 + 0.447i)$
$L(1)$  $\approx$  $1.877070953$
$L(\frac12)$  $\approx$  $1.877070953$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 - iT \)
good11 \( 1 - 4T + 11T^{2} \)
13 \( 1 + 3iT - 13T^{2} \)
17 \( 1 - 7iT - 17T^{2} \)
19 \( 1 - 6T + 19T^{2} \)
23 \( 1 + 9iT - 23T^{2} \)
29 \( 1 + 3T + 29T^{2} \)
31 \( 1 + 7T + 31T^{2} \)
37 \( 1 - 10iT - 37T^{2} \)
41 \( 1 + T + 41T^{2} \)
43 \( 1 - 13iT - 43T^{2} \)
47 \( 1 + 2iT - 47T^{2} \)
53 \( 1 - iT - 53T^{2} \)
59 \( 1 - 11T + 59T^{2} \)
61 \( 1 - 13T + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 - 8iT - 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 + 7iT - 83T^{2} \)
89 \( 1 - 14T + 89T^{2} \)
97 \( 1 + 8iT - 97T^{2} \)
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

−8.474084536017281232602483803504, −8.315633315099515390678892105422, −7.10726969544952645154602973998, −6.25433060652355187640828332068, −5.56798130957128575493504644335, −4.61089668806072020829715490227, −3.75260401564923922740596232305, −3.05351979585591314046285291365, −1.91802078772026257114788084570, −0.957964923863601312156335678334, 0.77860920818406676750222499911, 1.98513236476796832385261488948, 3.55955806459958725413863634078, 3.90985887244243575684573922447, 5.21194814994822844272447324282, 5.51999883862097348412744905372, 6.76911569919773997800164190505, 7.20572419112854146307497023922, 7.65607057226401001993405753570, 8.965519997760575876746827726754

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