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Properties

Label 2-135-9.4-c1-0-0
Degree $2$
Conductor $135$
Sign $-0.398 - 0.917i$
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  + (1.25 + 2.17i)2-s + (−2.16 + 3.74i)4-s + (0.5 − 0.866i)5-s + (0.257 + 0.445i)7-s − 5.83·8-s + 2.51·10-s + (−1.66 − 2.87i)11-s + (0.660 − 1.14i)13-s + (−0.646 + 1.11i)14-s + (−3.01 − 5.22i)16-s + 3.32·17-s − 1.32·19-s + (2.16 + 3.74i)20-s + (4.17 − 7.23i)22-s + (2.06 − 3.57i)23-s + ⋯
L(s)  = 1  + (0.888 + 1.53i)2-s + (−1.08 + 1.87i)4-s + (0.223 − 0.387i)5-s + (0.0971 + 0.168i)7-s − 2.06·8-s + 0.795·10-s + (−0.500 − 0.867i)11-s + (0.183 − 0.317i)13-s + (−0.172 + 0.299i)14-s + (−0.753 − 1.30i)16-s + 0.805·17-s − 0.303·19-s + (0.483 + 0.836i)20-s + (0.890 − 1.54i)22-s + (0.430 − 0.745i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.398 - 0.917i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.398 - 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(135\)    =    \(3^{3} \cdot 5\)
Sign: $-0.398 - 0.917i$
Analytic conductor: \(1.07798\)
Root analytic conductor: \(1.03825\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{135} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 135,\ (\ :1/2),\ -0.398 - 0.917i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.867707 + 1.32264i\)
\(L(\frac12)\) \(\approx\) \(0.867707 + 1.32264i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (-1.25 - 2.17i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + (-0.257 - 0.445i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.66 + 2.87i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.660 + 1.14i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 3.32T + 17T^{2} \)
19 \( 1 + 1.32T + 19T^{2} \)
23 \( 1 + (-2.06 + 3.57i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.693 + 1.20i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.36 - 7.56i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 0.292T + 37T^{2} \)
41 \( 1 + (5.67 - 9.82i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.17 + 8.96i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.43 + 4.21i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 5.02T + 53T^{2} \)
59 \( 1 + (2.51 - 4.35i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.67 + 6.36i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.72 - 8.18i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 8.99T + 71T^{2} \)
73 \( 1 - 6.05T + 73T^{2} \)
79 \( 1 + (-4.02 - 6.97i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.771 - 1.33i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 3T + 89T^{2} \)
97 \( 1 + (-6.12 - 10.6i)T + (-48.5 + 84.0i)T^{2} \)
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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

−13.64985302359493088093352985505, −12.99417451624685145059725651688, −11.99801402481293226192081675695, −10.43870046943605644397458072198, −8.781723923478140410568763495465, −8.124385896666467329514521756517, −6.87027369207621642519436798943, −5.73358348662922249710160424923, −4.96691338397767880240480542183, −3.40989333584331581688922123384, 1.88553473442488638822934478143, 3.31589077377514844905698331998, 4.60388041085956442777889062723, 5.79238938142523179117719186111, 7.46613653579566294475003308056, 9.317775776586409298204625515587, 10.17209842931830666640572120982, 11.00393264528283066524044130452, 11.92492703559235954525689083467, 12.85515096991660035134816487077

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