Properties

Label 2-165-5.4-c5-0-23
Degree $2$
Conductor $165$
Sign $0.770 - 0.637i$
Analytic cond. $26.4633$
Root an. cond. $5.14425$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.31i·2-s + 9i·3-s + 30.2·4-s + (−35.6 − 43.0i)5-s + 11.8·6-s + 87.4i·7-s − 82.0i·8-s − 81·9-s + (−56.7 + 46.9i)10-s + 121·11-s + 272. i·12-s + 521. i·13-s + 115.·14-s + (387. − 320. i)15-s + 860.·16-s + 68.1i·17-s + ⋯
L(s)  = 1  − 0.232i·2-s + 0.577i·3-s + 0.945·4-s + (−0.637 − 0.770i)5-s + 0.134·6-s + 0.674i·7-s − 0.453i·8-s − 0.333·9-s + (−0.179 + 0.148i)10-s + 0.301·11-s + 0.546i·12-s + 0.855i·13-s + 0.157·14-s + (0.444 − 0.368i)15-s + 0.840·16-s + 0.0571i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.770 - 0.637i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.770 - 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $0.770 - 0.637i$
Analytic conductor: \(26.4633\)
Root analytic conductor: \(5.14425\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{165} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 165,\ (\ :5/2),\ 0.770 - 0.637i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.192543319\)
\(L(\frac12)\) \(\approx\) \(2.192543319\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 9iT \)
5 \( 1 + (35.6 + 43.0i)T \)
11 \( 1 - 121T \)
good2 \( 1 + 1.31iT - 32T^{2} \)
7 \( 1 - 87.4iT - 1.68e4T^{2} \)
13 \( 1 - 521. iT - 3.71e5T^{2} \)
17 \( 1 - 68.1iT - 1.41e6T^{2} \)
19 \( 1 - 1.58e3T + 2.47e6T^{2} \)
23 \( 1 - 191. iT - 6.43e6T^{2} \)
29 \( 1 - 4.25e3T + 2.05e7T^{2} \)
31 \( 1 + 83.6T + 2.86e7T^{2} \)
37 \( 1 - 1.44e4iT - 6.93e7T^{2} \)
41 \( 1 - 6.15e3T + 1.15e8T^{2} \)
43 \( 1 - 1.40e3iT - 1.47e8T^{2} \)
47 \( 1 - 1.47e4iT - 2.29e8T^{2} \)
53 \( 1 - 1.09e4iT - 4.18e8T^{2} \)
59 \( 1 - 1.60e4T + 7.14e8T^{2} \)
61 \( 1 - 4.10e4T + 8.44e8T^{2} \)
67 \( 1 - 1.25e3iT - 1.35e9T^{2} \)
71 \( 1 - 4.76e4T + 1.80e9T^{2} \)
73 \( 1 + 7.35e3iT - 2.07e9T^{2} \)
79 \( 1 + 9.01e4T + 3.07e9T^{2} \)
83 \( 1 - 1.08e4iT - 3.93e9T^{2} \)
89 \( 1 + 1.12e5T + 5.58e9T^{2} \)
97 \( 1 + 9.69e4iT - 8.58e9T^{2} \)
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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

−11.77946589706944736433805433612, −11.38494247519712236322985062293, −10.00835770127684984731382630626, −9.060089461090847224560303118438, −8.009355869426122181597788600805, −6.72981538862580268093762608281, −5.44529258131571499011586771044, −4.17837098935044067764280194371, −2.86302567658525896242458274010, −1.25792157483903074419989700145, 0.802650503671775342022254272625, 2.52006836904914782789240651749, 3.67341697183289679044391147707, 5.61454416600208340396575571123, 6.83994847408242448081036458481, 7.38265970519279240186742975823, 8.281384428645557579022041489457, 10.09408981626091630481665034839, 10.96595571856152062826840855934, 11.72732120065056583200949159619

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