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Properties

Label 2-135-5.3-c2-0-13
Degree $2$
Conductor $135$
Sign $-0.359 + 0.933i$
Analytic cond. $3.67848$
Root an. cond. $1.91793$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.04 − 2.04i)2-s − 4.33i·4-s + (−4.12 − 2.81i)5-s + (8.42 − 8.42i)7-s + (−0.689 − 0.689i)8-s + (−14.1 + 2.67i)10-s − 12.2·11-s + (0.663 + 0.663i)13-s − 34.4i·14-s + 14.5·16-s + (7.47 − 7.47i)17-s + 25.0i·19-s + (−12.2 + 17.9i)20-s + (−25.1 + 25.1i)22-s + (18.1 + 18.1i)23-s + ⋯
L(s)  = 1  + (1.02 − 1.02i)2-s − 1.08i·4-s + (−0.825 − 0.563i)5-s + (1.20 − 1.20i)7-s + (−0.0861 − 0.0861i)8-s + (−1.41 + 0.267i)10-s − 1.11·11-s + (0.0510 + 0.0510i)13-s − 2.45i·14-s + 0.908·16-s + (0.439 − 0.439i)17-s + 1.31i·19-s + (−0.611 + 0.895i)20-s + (−1.14 + 1.14i)22-s + (0.787 + 0.787i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.359 + 0.933i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.359 + 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(135\)    =    \(3^{3} \cdot 5\)
Sign: $-0.359 + 0.933i$
Analytic conductor: \(3.67848\)
Root analytic conductor: \(1.91793\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{135} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 135,\ (\ :1),\ -0.359 + 0.933i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.26408 - 1.84073i\)
\(L(\frac12)\) \(\approx\) \(1.26408 - 1.84073i\)
\(L(2)\) 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 \( 1 + (4.12 + 2.81i)T \)
good2 \( 1 + (-2.04 + 2.04i)T - 4iT^{2} \)
7 \( 1 + (-8.42 + 8.42i)T - 49iT^{2} \)
11 \( 1 + 12.2T + 121T^{2} \)
13 \( 1 + (-0.663 - 0.663i)T + 169iT^{2} \)
17 \( 1 + (-7.47 + 7.47i)T - 289iT^{2} \)
19 \( 1 - 25.0iT - 361T^{2} \)
23 \( 1 + (-18.1 - 18.1i)T + 529iT^{2} \)
29 \( 1 - 3.38iT - 841T^{2} \)
31 \( 1 - 28.8T + 961T^{2} \)
37 \( 1 + (2.59 - 2.59i)T - 1.36e3iT^{2} \)
41 \( 1 + 23.3T + 1.68e3T^{2} \)
43 \( 1 + (11.6 + 11.6i)T + 1.84e3iT^{2} \)
47 \( 1 + (49.0 - 49.0i)T - 2.20e3iT^{2} \)
53 \( 1 + (-27.6 - 27.6i)T + 2.80e3iT^{2} \)
59 \( 1 + 87.7iT - 3.48e3T^{2} \)
61 \( 1 + 79.0T + 3.72e3T^{2} \)
67 \( 1 + (45.3 - 45.3i)T - 4.48e3iT^{2} \)
71 \( 1 - 88.7T + 5.04e3T^{2} \)
73 \( 1 + (-12.8 - 12.8i)T + 5.32e3iT^{2} \)
79 \( 1 + 30.3iT - 6.24e3T^{2} \)
83 \( 1 + (107. + 107. i)T + 6.88e3iT^{2} \)
89 \( 1 - 117. iT - 7.92e3T^{2} \)
97 \( 1 + (-13.5 + 13.5i)T - 9.40e3iT^{2} \)
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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

−12.63509412993166244725115635439, −11.68839349290911249676816642778, −10.99117139711257619174494101868, −10.11264226660286636541957379543, −8.151389754131920645478449893247, −7.56610637782276127600851665166, −5.24823856030604063576630098700, −4.50910427161865650702716397594, −3.37117710402993904560113996575, −1.34906801839293515729930892963, 2.84101635281826386558591341756, 4.60129879540445928469078869856, 5.36449286966380740817177754548, 6.68793953899344217712575933011, 7.82203614840333161620971973203, 8.530380861784489608914983407086, 10.49532976892008757348854377496, 11.52319317833180551070304998175, 12.45667145068145203182683875782, 13.50378547637176807825301633136

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