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Properties

Label 2-135-15.14-c2-0-7
Degree $2$
Conductor $135$
Sign $0.599 + 0.799i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.599 + 0.799i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.599 + 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(135\)    =    \(3^{3} \cdot 5\)
Sign: $0.599 + 0.799i$
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: \((2,\ 135,\ (\ :1),\ 0.599 + 0.799i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.925640 - 0.462820i\)
\(L(\frac12)\) \(\approx\) \(0.925640 - 0.462820i\)
\(L(2)\) 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

−13.01311446509182224511424580094, −11.89440559418743414493472113596, −10.48462576661018914831656225985, −9.599611856034358848112237736932, −8.636250793607475271511693499619, −8.013596345370321694350786965021, −5.79744115943471452123934807274, −5.31425940147799744845997758749, −3.24669887644390412664694550518, −0.954468687239003510964642111994, 1.69575349913410456652536905450, 3.94081590981531766672814753063, 5.21154962727406307687574987004, 6.94852542338980132184220431283, 7.66124022858928342547117465445, 9.431015022389270099359299302336, 9.783677443690611734661349983754, 10.71458093163368727443007528288, 12.21554048307745055700078555112, 13.27739047222339600482843778846

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