Properties

Label 2-285-15.8-c1-0-25
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

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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: $\chi_{285} (248, \cdot )$
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 \( 1 + (-7.14 + 7.14i)T - 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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