Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $-0.447 + 0.894i$
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 − 6i·13-s − 14-s + 16-s − 2i·17-s + 4·19-s − 4i·22-s + 8i·23-s − 6·26-s + i·28-s − 2·29-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s − 0.377i·7-s + 0.353i·8-s + 1.20·11-s − 1.66i·13-s − 0.267·14-s + 0.250·16-s − 0.485i·17-s + 0.917·19-s − 0.852i·22-s + 1.66i·23-s − 1.17·26-s + 0.188i·28-s − 0.371·29-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.447 + 0.894i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (2899, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ -0.447 + 0.894i)$
$L(1)$  $\approx$  $1.794713349$
$L(\frac12)$  $\approx$  $1.794713349$
$L(\frac{3}{2})$   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 \( 1 + 6iT - 13T^{2} \)
17 \( 1 + 2iT - 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 - 8iT - 23T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 10iT - 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 - 6T + 61T^{2} \)
67 \( 1 - 4iT - 67T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + 10iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 4iT - 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + 14iT - 97T^{2} \)
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

−8.575688041178342675376580720894, −7.58708883784009702265886652514, −7.24352896766114055981756508196, −5.88122106670547835414241782326, −5.44936849187633603931885850179, −4.34336573609305764327316516822, −3.54646280687156728472762582249, −2.91013921001833239425327250775, −1.56770016999091544996877878190, −0.65489759606315591206840315201, 1.18213511157640488656663341229, 2.31982376803787028489023508474, 3.65610088890848707296211559053, 4.32895092309883762214038376186, 5.10187447989534779390674500199, 6.18226285667188940045780795352, 6.59696200645346197085099445930, 7.24146641322850673576644518077, 8.310515878353982249337371850513, 8.825662352621021287999239670911

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