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Properties

Label 2-135-15.8-c1-0-4
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

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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: $\chi_{135} (53, \cdot )$
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 \( 1 + (3.58 - 3.58i)T - 97iT^{2} \)
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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

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

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