↑

Properties

Label 2-945-315.139-c0-0-0
Degree $2$
Conductor $945$
Sign $-0.642 - 0.766i$
Analytic cond. $0.471616$
Root an. cond. $0.686743$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)4-s + (−0.5 + 0.866i)5-s + (0.5 + 0.866i)7-s + (−0.5 − 0.866i)11-s + (−0.5 + 0.866i)13-s + (−0.499 − 0.866i)16-s − 17-s + (−0.499 − 0.866i)20-s + (−0.499 − 0.866i)25-s − 0.999·28-s + (1 + 1.73i)29-s − 0.999·35-s + 0.999·44-s + (0.5 + 0.866i)47-s + (−0.499 + 0.866i)49-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)4-s + (−0.5 + 0.866i)5-s + (0.5 + 0.866i)7-s + (−0.5 − 0.866i)11-s + (−0.5 + 0.866i)13-s + (−0.499 − 0.866i)16-s − 17-s + (−0.499 − 0.866i)20-s + (−0.499 − 0.866i)25-s − 0.999·28-s + (1 + 1.73i)29-s − 0.999·35-s + 0.999·44-s + (0.5 + 0.866i)47-s + (−0.499 + 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(945\)    =    \(3^{3} \cdot 5 \cdot 7\)
Sign: $-0.642 - 0.766i$
Analytic conductor: \(0.471616\)
Root analytic conductor: \(0.686743\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{945} (874, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 945,\ (\ :0),\ -0.642 - 0.766i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6809200523\)
\(L(\frac12)\) \(\approx\) \(0.6809200523\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + T + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 - T + T^{2} \)
73 \( 1 - T + T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.82917599296387125745317909102, −9.532782310488458177315716868608, −8.695906947112781539843189901559, −8.192409724254205467234083184664, −7.23615400486247066909131052547, −6.46014357494292055649862723112, −5.17572676929868189460097534854, −4.30860543349632076640617416905, −3.19163578160091161782215306127, −2.38865482046708246309519136320, 0.66210645844567043654430093127, 2.13296047652412830308913082438, 4.00835789159501541687384154746, 4.70674051537928115428115562533, 5.27029414542421131394738961681, 6.52102065956943462282851621884, 7.62295267227576626149145311636, 8.214437140811042854537579808206, 9.186666978843783051389022427274, 10.02974132397202169235033174427

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