Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $0.106 + 0.994i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.106 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.106 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.106 + 0.994i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (1151, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ 0.106 + 0.994i)$
$L(1)$  $\approx$  $2.241157938$
$L(\frac12)$  $\approx$  $2.241157938$
$L(\frac{3}{2})$   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 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 + (2.63 - 0.189i)T \)
good11 \( 1 + (-1.32 - 0.766i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 1.48iT - 13T^{2} \)
17 \( 1 + (-1.21 + 2.10i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.21 + 2.43i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.232 + 0.133i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 0.898iT - 29T^{2} \)
31 \( 1 + (0.717 + 0.414i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.74 + 4.74i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 8.76T + 41T^{2} \)
43 \( 1 + 1.86T + 43T^{2} \)
47 \( 1 + (3.72 + 6.46i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.00 - 1.73i)T + (26.5 + 45.8i)T^{2} \)
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 \( 1 + (-8.01 + 13.8i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 12.7iT - 71T^{2} \)
73 \( 1 + (0.297 + 0.171i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.22 - 9.04i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 5.45T + 83T^{2} \)
89 \( 1 + (7.98 + 13.8i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 14.9iT - 97T^{2} \)
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

−8.707303671293084546820644773816, −7.47349077189403169030741138811, −6.95902362333323240469039567192, −6.16526415323563916159170994196, −5.42643607108847155540659925252, −4.56874911847277851574686297749, −3.66711927473721145835515703782, −2.99288672112097463071876877885, −1.99912987424262084346938670304, −0.62639867694176628327600978419, 1.14033892577187205122942301716, 2.62850871805637710856031411042, 3.44298153663581433000624997825, 4.02950002662229611592227559352, 5.18620478694166003688623136852, 5.84011636557687166306493860596, 6.50617504843627434620667080398, 7.23094348931081179965012105444, 7.991293596277286815653237023404, 8.757268326017797568598487568035

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