Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $0.656 + 0.754i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + (2.23 − 1.41i)7-s + 8-s + 1.41i·11-s − 0.926·13-s + (2.23 − 1.41i)14-s + 16-s − 2.23i·17-s − 7.63i·19-s + 1.41i·22-s + 23-s − 0.926·26-s + (2.23 − 1.41i)28-s − 0.757i·29-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + (0.845 − 0.534i)7-s + 0.353·8-s + 0.426i·11-s − 0.256·13-s + (0.597 − 0.377i)14-s + 0.250·16-s − 0.542i·17-s − 1.75i·19-s + 0.301i·22-s + 0.208·23-s − 0.181·26-s + (0.422 − 0.267i)28-s − 0.140i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.656 + 0.754i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.656 + 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.656 + 0.754i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (3149, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ 0.656 + 0.754i)$
$L(1)$  $\approx$  $3.192398685$
$L(\frac12)$  $\approx$  $3.192398685$
$L(\frac{3}{2})$   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 \( 1 - T \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 + (-2.23 + 1.41i)T \)
good11 \( 1 - 1.41iT - 11T^{2} \)
13 \( 1 + 0.926T + 13T^{2} \)
17 \( 1 + 2.23iT - 17T^{2} \)
19 \( 1 + 7.63iT - 19T^{2} \)
23 \( 1 - T + 23T^{2} \)
29 \( 1 + 0.757iT - 29T^{2} \)
31 \( 1 + 4.08iT - 31T^{2} \)
37 \( 1 - 2.82iT - 37T^{2} \)
41 \( 1 - 8.56T + 41T^{2} \)
43 \( 1 + 3.58iT - 43T^{2} \)
47 \( 1 - 1.30iT - 47T^{2} \)
53 \( 1 - 8.07T + 53T^{2} \)
59 \( 1 + 7.25T + 59T^{2} \)
61 \( 1 + 0.926iT - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 - 15.6iT - 71T^{2} \)
73 \( 1 + 13.9T + 73T^{2} \)
79 \( 1 - 13.0T + 79T^{2} \)
83 \( 1 + 14.3iT - 83T^{2} \)
89 \( 1 - 2.61T + 89T^{2} \)
97 \( 1 - 0.542T + 97T^{2} \)
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

−8.543564384582761225793051565292, −7.44043419365128599530770341839, −7.27427504373926763259451677054, −6.29098405693889517259993328045, −5.30652162313399700602993636286, −4.67053212285144311455243651758, −4.11860257275618043111956043719, −2.90746544979422295273135663843, −2.12094465379866934711623394134, −0.811350011978743316061453134382, 1.34991033559655165765495886466, 2.24992848327993913588066033395, 3.30265025529125945214350900673, 4.13336154770751772102268930643, 4.96843829142805050237046161272, 5.75241252348345247993384919327, 6.21567423046819211608926096830, 7.35735014431849800587988288287, 7.972218724067222347796060461253, 8.632496792650311944668362068227

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