Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $-0.641 - 0.766i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (0.189 − 2.63i)7-s − 0.999·8-s + (−1.32 + 0.766i)11-s − 1.48·13-s + (−2.19 − 1.48i)14-s + (−0.5 + 0.866i)16-s + (2.10 − 1.21i)17-s + (−4.21 − 2.43i)19-s + 1.53i·22-s + (−0.133 + 0.232i)23-s + (−0.741 + 1.28i)26-s + (−2.38 + 1.15i)28-s + 0.898i·29-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.0716 − 0.997i)7-s − 0.353·8-s + (−0.400 + 0.230i)11-s − 0.411·13-s + (−0.585 − 0.396i)14-s + (−0.125 + 0.216i)16-s + (0.510 − 0.294i)17-s + (−0.966 − 0.557i)19-s + 0.326i·22-s + (−0.0279 + 0.0483i)23-s + (−0.145 + 0.251i)26-s + (−0.449 + 0.218i)28-s + 0.166i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.641 - 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.641 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.641 - 0.766i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (1349, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ -0.641 - 0.766i)$
$L(1)$  $\approx$  $0.4757454367$
$L(\frac12)$  $\approx$  $0.4757454367$
$L(\frac{3}{2})$   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 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 + (-0.189 + 2.63i)T \)
good11 \( 1 + (1.32 - 0.766i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 1.48T + 13T^{2} \)
17 \( 1 + (-2.10 + 1.21i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.21 + 2.43i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.133 - 0.232i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 0.898iT - 29T^{2} \)
31 \( 1 + (0.717 - 0.414i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.74 + 2.74i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 8.76T + 41T^{2} \)
43 \( 1 - 1.86iT - 43T^{2} \)
47 \( 1 + (-6.46 - 3.72i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.73 - 3.00i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.12 + 5.41i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.73 - 3.31i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (13.8 - 8.01i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 12.7iT - 71T^{2} \)
73 \( 1 + (-0.171 - 0.297i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.22 - 9.04i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 5.45iT - 83T^{2} \)
89 \( 1 + (7.98 - 13.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 14.9T + 97T^{2} \)
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

−8.235466744752119998768525828149, −7.34770265706349466822307633002, −6.80939389258504457924637868804, −5.76272821286160704832224618727, −4.90701777014296741962000139728, −4.28808298300726631902198736517, −3.41024905765645980818057340139, −2.49174462520679842273035628936, −1.40986879871683962213407304302, −0.12405743037466701370456116708, 1.82299298221926028083235423501, 2.81143098735057610283676889671, 3.72483544017149815098422462632, 4.71662522717045056673386692317, 5.47699312942237990194355447968, 6.01027431100877783281277140211, 6.84549564635507455216753937968, 7.66582443705115884985033693934, 8.488983911449326369152498948632, 8.771792129853909060920662610966

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