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Properties

Label 2-135-5.4-c1-0-2
Degree $2$
Conductor $135$
Sign $0.316 - 0.948i$
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  + 1.41i·2-s + (2.12 + 0.707i)5-s − 3i·7-s + 2.82i·8-s + (−1.00 + 3i)10-s − 4.24·11-s + 3i·13-s + 4.24·14-s − 4.00·16-s − 2.82i·17-s − 19-s − 6i·22-s − 7.07i·23-s + (3.99 + 3i)25-s − 4.24·26-s + ⋯
L(s)  = 1  + 0.999i·2-s + (0.948 + 0.316i)5-s − 1.13i·7-s + 0.999i·8-s + (−0.316 + 0.948i)10-s − 1.27·11-s + 0.832i·13-s + 1.13·14-s − 1.00·16-s − 0.685i·17-s − 0.229·19-s − 1.27i·22-s − 1.47i·23-s + (0.799 + 0.600i)25-s − 0.832·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.316 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.316 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(135\)    =    \(3^{3} \cdot 5\)
Sign: $0.316 - 0.948i$
Analytic conductor: \(1.07798\)
Root analytic conductor: \(1.03825\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{135} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 135,\ (\ :1/2),\ 0.316 - 0.948i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03514 + 0.746092i\)
\(L(\frac12)\) \(\approx\) \(1.03514 + 0.746092i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.12 - 0.707i)T \)
good2 \( 1 - 1.41iT - 2T^{2} \)
7 \( 1 + 3iT - 7T^{2} \)
11 \( 1 + 4.24T + 11T^{2} \)
13 \( 1 - 3iT - 13T^{2} \)
17 \( 1 + 2.82iT - 17T^{2} \)
19 \( 1 + T + 19T^{2} \)
23 \( 1 + 7.07iT - 23T^{2} \)
29 \( 1 + 4.24T + 29T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 + 9iT - 37T^{2} \)
41 \( 1 - 4.24T + 41T^{2} \)
43 \( 1 - 6iT - 43T^{2} \)
47 \( 1 + 2.82iT - 47T^{2} \)
53 \( 1 - 9.89iT - 53T^{2} \)
59 \( 1 + 8.48T + 59T^{2} \)
61 \( 1 + 13T + 61T^{2} \)
67 \( 1 - 3iT - 67T^{2} \)
71 \( 1 - 12.7T + 71T^{2} \)
73 \( 1 - 9iT - 73T^{2} \)
79 \( 1 - 5T + 79T^{2} \)
83 \( 1 - 1.41iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + 3iT - 97T^{2} \)
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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

−13.85174070080401332303088791745, −12.74279709354293528097188993695, −11.06894559519364354007467995035, −10.45096784905835340771338318157, −9.158001478947800703851831006164, −7.75017696126545425844621574372, −6.95737592293525677656342898956, −5.95031934677130676969642251552, −4.68698523821464655812134364779, −2.45660321617197241438548096896, 1.93981026467338891151854269245, 3.08130577747486131419126422490, 5.20419041329673760118541883973, 6.14522138017574112019349301678, 7.896057077059508667919054658929, 9.192806302771851579721409368435, 10.10161790890457888504275628339, 10.91599625559941872878930190643, 12.11715466140420903553210589085, 12.87844915165123163353345812929

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