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Properties

Label 2-135-15.14-c2-0-15
Degree $2$
Conductor $135$
Sign $-0.599 + 0.800i$
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-s − 3·4-s + (−4 − 3i)5-s − 6i·7-s − 7·8-s + (−4 − 3i)10-s − 21i·11-s + 15i·13-s − 6i·14-s + 5·16-s − 23·17-s + 14·19-s + (12 + 9i)20-s − 21i·22-s + 7·23-s + ⋯
L(s)  = 1  + 0.5·2-s − 0.750·4-s + (−0.800 − 0.600i)5-s − 0.857i·7-s − 0.875·8-s + (−0.400 − 0.300i)10-s − 1.90i·11-s + 1.15i·13-s − 0.428i·14-s + 0.312·16-s − 1.35·17-s + 0.736·19-s + (0.600 + 0.450i)20-s − 0.954i·22-s + 0.304·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.599 + 0.800i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.599 + 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(135\)    =    \(3^{3} \cdot 5\)
Sign: $-0.599 + 0.800i$
Analytic conductor: \(3.67848\)
Root analytic conductor: \(1.91793\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{135} (134, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 135,\ (\ :1),\ -0.599 + 0.800i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.378462 - 0.756924i\)
\(L(\frac12)\) \(\approx\) \(0.378462 - 0.756924i\)
\(L(2)\) 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 \( 1 + (4 + 3i)T \)
good2 \( 1 - T + 4T^{2} \)
7 \( 1 + 6iT - 49T^{2} \)
11 \( 1 + 21iT - 121T^{2} \)
13 \( 1 - 15iT - 169T^{2} \)
17 \( 1 + 23T + 289T^{2} \)
19 \( 1 - 14T + 361T^{2} \)
23 \( 1 - 7T + 529T^{2} \)
29 \( 1 + 3iT - 841T^{2} \)
31 \( 1 + 25T + 961T^{2} \)
37 \( 1 + 54iT - 1.36e3T^{2} \)
41 \( 1 + 24iT - 1.68e3T^{2} \)
43 \( 1 + 15iT - 1.84e3T^{2} \)
47 \( 1 - 49T + 2.20e3T^{2} \)
53 \( 1 + 14T + 2.80e3T^{2} \)
59 \( 1 - 30iT - 3.48e3T^{2} \)
61 \( 1 - 44T + 3.72e3T^{2} \)
67 \( 1 + 66iT - 4.48e3T^{2} \)
71 \( 1 + 18iT - 5.04e3T^{2} \)
73 \( 1 - 5.32e3T^{2} \)
79 \( 1 + 37T + 6.24e3T^{2} \)
83 \( 1 + 116T + 6.88e3T^{2} \)
89 \( 1 + 126iT - 7.92e3T^{2} \)
97 \( 1 + 78iT - 9.40e3T^{2} \)
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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

−12.85331044641990316121117522697, −11.63515626638235820242496552235, −10.89420486992489271880015721999, −9.136173722178369989343209508975, −8.630452659139887399070286407169, −7.23009251612438383257197774072, −5.71036455748717903727887887958, −4.38432179778500552481414062198, −3.58737852672479937214642523246, −0.49733648887173635979280610102, 2.78324299508896463918830938374, 4.26523031623135017566182515384, 5.30223351582427962162883624061, 6.83856738503115887732123229852, 8.029568490297505460588325771580, 9.194096736926393775897213053717, 10.23106262575648244505316990241, 11.60036849916936983577732386121, 12.46647950387815535738980075041, 13.14909080070799417239053923887

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