Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $-0.503 + 0.863i$
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 + (−2.64 − 0.0951i)7-s + 8-s + 5.28i·11-s − 2.19·13-s + (−2.64 − 0.0951i)14-s + 16-s − 1.04i·17-s − 6.43i·19-s + 5.28i·22-s − 7.47·23-s − 2.19·26-s + (−2.64 − 0.0951i)28-s − 7.47i·29-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + (−0.999 − 0.0359i)7-s + 0.353·8-s + 1.59i·11-s − 0.607·13-s + (−0.706 − 0.0254i)14-s + 0.250·16-s − 0.253i·17-s − 1.47i·19-s + 1.12i·22-s − 1.55·23-s − 0.429·26-s + (−0.499 − 0.0179i)28-s − 1.38i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.503 + 0.863i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.503 + 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.503 + 0.863i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (3149, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ -0.503 + 0.863i)$
$L(1)$  $\approx$  $1.086077824$
$L(\frac12)$  $\approx$  $1.086077824$
$L(\frac{3}{2})$   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 \( 1 + (2.64 + 0.0951i)T \)
good11 \( 1 - 5.28iT - 11T^{2} \)
13 \( 1 + 2.19T + 13T^{2} \)
17 \( 1 + 1.04iT - 17T^{2} \)
19 \( 1 + 6.43iT - 19T^{2} \)
23 \( 1 + 7.47T + 23T^{2} \)
29 \( 1 + 7.47iT - 29T^{2} \)
31 \( 1 + 9.09iT - 31T^{2} \)
37 \( 1 - 0.855iT - 37T^{2} \)
41 \( 1 + 2.19T + 41T^{2} \)
43 \( 1 + 0.954iT - 43T^{2} \)
47 \( 1 + 11.0iT - 47T^{2} \)
53 \( 1 - 3.09T + 53T^{2} \)
59 \( 1 - 13.7T + 59T^{2} \)
61 \( 1 - 8.05iT - 61T^{2} \)
67 \( 1 + 5.33iT - 67T^{2} \)
71 \( 1 + 6.43iT - 71T^{2} \)
73 \( 1 + 4.57T + 73T^{2} \)
79 \( 1 + 15.6T + 79T^{2} \)
83 \( 1 - 4.38iT - 83T^{2} \)
89 \( 1 - 4.28T + 89T^{2} \)
97 \( 1 + 11.8T + 97T^{2} \)
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

−8.322083192810818460354717705106, −7.32792033151827832735832527101, −6.99277901574765661101529115608, −6.14389298626220996784144724046, −5.34082685368333877435514141345, −4.39253173457436890138673287534, −3.93142246593743288893567514538, −2.60353331875382842459920240141, −2.15828539523710888270885430788, −0.24417949452123263842519559640, 1.40105246598310387289546053170, 2.73353709572955696369996338504, 3.44741772341814301699968409238, 4.04188764774939784795959450698, 5.30449143489844164978353212323, 5.86226434260004731243847136573, 6.47178123312405743660240213412, 7.27822501348135362951924994664, 8.234245956610141589294581188966, 8.766421612219231158240890194025

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