Properties

Label 2-285-15.2-c1-0-15
Degree $2$
Conductor $285$
Sign $0.990 + 0.137i$
Analytic cond. $2.27573$
Root an. cond. $1.50855$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.60 − 1.60i)2-s + (−0.918 + 1.46i)3-s − 3.16i·4-s + (−0.294 + 2.21i)5-s + (0.884 + 3.83i)6-s + (3.56 + 3.56i)7-s + (−1.87 − 1.87i)8-s + (−1.31 − 2.69i)9-s + (3.08 + 4.03i)10-s − 3.54i·11-s + (4.65 + 2.90i)12-s + (−1.01 + 1.01i)13-s + 11.4·14-s + (−2.98 − 2.46i)15-s + 0.299·16-s + (−0.0645 + 0.0645i)17-s + ⋯
L(s)  = 1  + (1.13 − 1.13i)2-s + (−0.530 + 0.847i)3-s − 1.58i·4-s + (−0.131 + 0.991i)5-s + (0.360 + 1.56i)6-s + (1.34 + 1.34i)7-s + (−0.663 − 0.663i)8-s + (−0.437 − 0.899i)9-s + (0.976 + 1.27i)10-s − 1.06i·11-s + (1.34 + 0.839i)12-s + (−0.282 + 0.282i)13-s + 3.06·14-s + (−0.770 − 0.637i)15-s + 0.0749·16-s + (−0.0156 + 0.0156i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.137i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 + 0.137i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(285\)    =    \(3 \cdot 5 \cdot 19\)
Sign: $0.990 + 0.137i$
Analytic conductor: \(2.27573\)
Root analytic conductor: \(1.50855\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{285} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 285,\ (\ :1/2),\ 0.990 + 0.137i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.04860 - 0.141159i\)
\(L(\frac12)\) \(\approx\) \(2.04860 - 0.141159i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.918 - 1.46i)T \)
5 \( 1 + (0.294 - 2.21i)T \)
19 \( 1 + iT \)
good2 \( 1 + (-1.60 + 1.60i)T - 2iT^{2} \)
7 \( 1 + (-3.56 - 3.56i)T + 7iT^{2} \)
11 \( 1 + 3.54iT - 11T^{2} \)
13 \( 1 + (1.01 - 1.01i)T - 13iT^{2} \)
17 \( 1 + (0.0645 - 0.0645i)T - 17iT^{2} \)
23 \( 1 + (3.09 + 3.09i)T + 23iT^{2} \)
29 \( 1 - 0.873T + 29T^{2} \)
31 \( 1 + 1.93T + 31T^{2} \)
37 \( 1 + (-1.47 - 1.47i)T + 37iT^{2} \)
41 \( 1 + 8.53iT - 41T^{2} \)
43 \( 1 + (4.61 - 4.61i)T - 43iT^{2} \)
47 \( 1 + (-4.86 + 4.86i)T - 47iT^{2} \)
53 \( 1 + (-0.315 - 0.315i)T + 53iT^{2} \)
59 \( 1 + 13.1T + 59T^{2} \)
61 \( 1 - 5.71T + 61T^{2} \)
67 \( 1 + (8.04 + 8.04i)T + 67iT^{2} \)
71 \( 1 + 5.53iT - 71T^{2} \)
73 \( 1 + (-9.31 + 9.31i)T - 73iT^{2} \)
79 \( 1 + 0.183iT - 79T^{2} \)
83 \( 1 + (-2.39 - 2.39i)T + 83iT^{2} \)
89 \( 1 + 10.4T + 89T^{2} \)
97 \( 1 + (-7.14 - 7.14i)T + 97iT^{2} \)
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

−11.87734930624892608476675064852, −10.95876917576373011974519327938, −10.64564328935255981742345739480, −9.291104544034503726087570364527, −8.141712175249713758967192363796, −6.21330835027146856808638899650, −5.43767679252637717287049703907, −4.51412782364324367320543492174, −3.32444160317024225709481553409, −2.23684677695064981649078722054, 1.47880360254987113348339329275, 4.19340864176303194146699370723, 4.82680221584479806379518158616, 5.67348259292418041047969138053, 7.03279234086989888231309687978, 7.64298913697956770895473235794, 8.227069673185028087383802568339, 10.08420821298545191330812886096, 11.34773390474554701206858870205, 12.22417260105583685758250157721

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