Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $0.924 - 0.381i$
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 + (1.41 − 2.23i)7-s + i·8-s − 1.41i·11-s + 5.39i·13-s + (−2.23 − 1.41i)14-s + 16-s − 2.23·17-s + 1.30i·19-s − 1.41·22-s + i·23-s + 5.39·26-s + (−1.41 + 2.23i)28-s + 9.24i·29-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + (0.534 − 0.845i)7-s + 0.353i·8-s − 0.426i·11-s + 1.49i·13-s + (−0.597 − 0.377i)14-s + 0.250·16-s − 0.542·17-s + 0.300i·19-s − 0.301·22-s + 0.208i·23-s + 1.05·26-s + (−0.267 + 0.422i)28-s + 1.71i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.924 - 0.381i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.924 - 0.381i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.924 - 0.381i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (251, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ 0.924 - 0.381i)$
$L(1)$  $\approx$  $1.382635125$
$L(\frac12)$  $\approx$  $1.382635125$
$L(\frac{3}{2})$   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 \( 1 + (-1.41 + 2.23i)T \)
good11 \( 1 + 1.41iT - 11T^{2} \)
13 \( 1 - 5.39iT - 13T^{2} \)
17 \( 1 + 2.23T + 17T^{2} \)
19 \( 1 - 1.30iT - 19T^{2} \)
23 \( 1 - iT - 23T^{2} \)
29 \( 1 - 9.24iT - 29T^{2} \)
31 \( 1 - 8.56iT - 31T^{2} \)
37 \( 1 + 2.82T + 37T^{2} \)
41 \( 1 + 4.08T + 41T^{2} \)
43 \( 1 - 6.41T + 43T^{2} \)
47 \( 1 - 7.63T + 47T^{2} \)
53 \( 1 + 6.07iT - 53T^{2} \)
59 \( 1 + 11.7T + 59T^{2} \)
61 \( 1 - 5.39iT - 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 - 4.34iT - 71T^{2} \)
73 \( 1 - 5.01iT - 73T^{2} \)
79 \( 1 - 1.07T + 79T^{2} \)
83 \( 1 - 8.01T + 83T^{2} \)
89 \( 1 + 15.2T + 89T^{2} \)
97 \( 1 - 18.4iT - 97T^{2} \)
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

−8.889038775068790750227531325626, −8.157252257078707806810864205516, −7.12559550820503319378205254159, −6.69273094183610470677740076247, −5.46241566945439418216833661152, −4.67978600963685885508351668481, −3.99136114074632710822619499417, −3.19426032668186898644493550524, −1.94017469584060185662696472561, −1.18189696386556021624067733787, 0.46153211081304693235084636242, 2.06679718276531990216365453631, 2.94825695922640576597935204213, 4.20633704491753260039139655677, 4.86671231669597730127577326025, 5.81027650091999692218304718117, 6.10959491672010731529697508472, 7.33851656520017785001363581100, 7.84383843387732832754600959707, 8.486487194277671848919392272147

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