↑

Properties

Label 2-135-15.2-c1-0-3
Degree $2$
Conductor $135$
Sign $0.725 - 0.688i$
Analytic cond. $1.07798$
Root an. cond. $1.03825$
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  + (−0.323 + 0.323i)2-s + 1.79i·4-s + (1.22 − 1.87i)5-s + (1.79 + 1.79i)7-s + (−1.22 − 1.22i)8-s + (0.208 + 0.999i)10-s + 5.54i·11-s + (1.79 − 1.79i)13-s − 1.15·14-s − 2.79·16-s + (1.87 − 1.87i)17-s − 3i·19-s + (3.35 + 2.19i)20-s + (−1.79 − 1.79i)22-s + (−4.32 − 4.32i)23-s + ⋯
L(s)  = 1  + (−0.228 + 0.228i)2-s + 0.895i·4-s + (0.547 − 0.836i)5-s + (0.677 + 0.677i)7-s + (−0.433 − 0.433i)8-s + (0.0660 + 0.316i)10-s + 1.67i·11-s + (0.496 − 0.496i)13-s − 0.309·14-s − 0.697·16-s + (0.453 − 0.453i)17-s − 0.688i·19-s + (0.749 + 0.490i)20-s + (−0.381 − 0.381i)22-s + (−0.900 − 0.900i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.725 - 0.688i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.725 - 0.688i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(135\)    =    \(3^{3} \cdot 5\)
Sign: $0.725 - 0.688i$
Analytic conductor: \(1.07798\)
Root analytic conductor: \(1.03825\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{135} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 135,\ (\ :1/2),\ 0.725 - 0.688i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00486 + 0.400969i\)
\(L(\frac12)\) \(\approx\) \(1.00486 + 0.400969i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (-1.22 + 1.87i)T \)
good2 \( 1 + (0.323 - 0.323i)T - 2iT^{2} \)
7 \( 1 + (-1.79 - 1.79i)T + 7iT^{2} \)
11 \( 1 - 5.54iT - 11T^{2} \)
13 \( 1 + (-1.79 + 1.79i)T - 13iT^{2} \)
17 \( 1 + (-1.87 + 1.87i)T - 17iT^{2} \)
19 \( 1 + 3iT - 19T^{2} \)
23 \( 1 + (4.32 + 4.32i)T + 23iT^{2} \)
29 \( 1 + 4.38T + 29T^{2} \)
31 \( 1 + T + 31T^{2} \)
37 \( 1 + (5 + 5i)T + 37iT^{2} \)
41 \( 1 - 5.54iT - 41T^{2} \)
43 \( 1 + (-3.20 + 3.20i)T - 43iT^{2} \)
47 \( 1 + (-0.646 + 0.646i)T - 47iT^{2} \)
53 \( 1 + (-0.0674 - 0.0674i)T + 53iT^{2} \)
59 \( 1 - 4.38T + 59T^{2} \)
61 \( 1 + T + 61T^{2} \)
67 \( 1 + (-8.58 - 8.58i)T + 67iT^{2} \)
71 \( 1 - 5.54iT - 71T^{2} \)
73 \( 1 + (10.3 - 10.3i)T - 73iT^{2} \)
79 \( 1 - 10.5iT - 79T^{2} \)
83 \( 1 + (8.70 + 8.70i)T + 83iT^{2} \)
89 \( 1 - 16.6T + 89T^{2} \)
97 \( 1 + (3.58 + 3.58i)T + 97iT^{2} \)
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

−13.03854446736934493498721445837, −12.46846489737220114243643761998, −11.61609993281928945235842966793, −9.997577919320859843866702623095, −9.008138393135755701146257930127, −8.181330501512705467690782992552, −7.07657552102532176137596332465, −5.49023505411011861881754541121, −4.31495365375244826910554547607, −2.23795266877774120938677657403, 1.61713194195920318200066816582, 3.59922831176414174465889338660, 5.56147505710240121737893587048, 6.31806941393255689084706515881, 7.83470481704393807457943601585, 9.102594990326131027285332293248, 10.28138605531598066743605464214, 10.90185203004764225257335910539, 11.64145880607481508308750588512, 13.61489495128882556220892525395

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