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Properties

Label 2-15-3.2-c10-0-11
Degree $2$
Conductor $15$
Sign $-0.944 - 0.329i$
Analytic cond. $9.53035$
Root an. cond. $3.08712$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 27.5i·2-s + (−80.1 + 229. i)3-s + 266.·4-s − 1.39e3i·5-s + (6.31e3 + 2.20e3i)6-s − 2.41e4·7-s − 3.55e4i·8-s + (−4.62e4 − 3.67e4i)9-s − 3.84e4·10-s + 2.18e5i·11-s + (−2.13e4 + 6.11e4i)12-s − 4.35e5·13-s + 6.63e5i·14-s + (3.20e5 + 1.11e5i)15-s − 7.04e5·16-s − 1.40e6i·17-s + ⋯
L(s)  = 1  − 0.860i·2-s + (−0.329 + 0.944i)3-s + 0.260·4-s − 0.447i·5-s + (0.812 + 0.283i)6-s − 1.43·7-s − 1.08i·8-s + (−0.782 − 0.622i)9-s − 0.384·10-s + 1.35i·11-s + (−0.0857 + 0.245i)12-s − 1.17·13-s + 1.23i·14-s + (0.422 + 0.147i)15-s − 0.672·16-s − 0.987i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 - 0.329i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.944 - 0.329i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $-0.944 - 0.329i$
Analytic conductor: \(9.53035\)
Root analytic conductor: \(3.08712\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{15} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :5),\ -0.944 - 0.329i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.0244569 + 0.144167i\)
\(L(\frac12)\) \(\approx\) \(0.0244569 + 0.144167i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (80.1 - 229. i)T \)
5 \( 1 + 1.39e3iT \)
good2 \( 1 + 27.5iT - 1.02e3T^{2} \)
7 \( 1 + 2.41e4T + 2.82e8T^{2} \)
11 \( 1 - 2.18e5iT - 2.59e10T^{2} \)
13 \( 1 + 4.35e5T + 1.37e11T^{2} \)
17 \( 1 + 1.40e6iT - 2.01e12T^{2} \)
19 \( 1 + 3.96e6T + 6.13e12T^{2} \)
23 \( 1 - 1.21e6iT - 4.14e13T^{2} \)
29 \( 1 - 1.00e6iT - 4.20e14T^{2} \)
31 \( 1 - 5.09e7T + 8.19e14T^{2} \)
37 \( 1 - 9.23e6T + 4.80e15T^{2} \)
41 \( 1 - 4.72e7iT - 1.34e16T^{2} \)
43 \( 1 + 7.55e7T + 2.16e16T^{2} \)
47 \( 1 - 1.53e8iT - 5.25e16T^{2} \)
53 \( 1 + 3.89e8iT - 1.74e17T^{2} \)
59 \( 1 + 1.11e8iT - 5.11e17T^{2} \)
61 \( 1 + 6.20e7T + 7.13e17T^{2} \)
67 \( 1 + 8.74e8T + 1.82e18T^{2} \)
71 \( 1 - 1.83e9iT - 3.25e18T^{2} \)
73 \( 1 - 7.05e7T + 4.29e18T^{2} \)
79 \( 1 + 6.40e8T + 9.46e18T^{2} \)
83 \( 1 + 3.98e9iT - 1.55e19T^{2} \)
89 \( 1 + 3.01e9iT - 3.11e19T^{2} \)
97 \( 1 + 1.02e10T + 7.37e19T^{2} \)
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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

−16.13121069497150620504408879589, −15.12953650272164850413762004439, −12.80698333914589289962308794132, −11.90895313951115889311942625302, −10.15416596796600595629569011360, −9.555998417491417772059939405469, −6.67251737975134337904639589816, −4.46026503320207301253530707196, −2.69571767982127780046997197040, −0.06548817583253833479184203277, 2.64708061569394358171107164621, 6.03227908636252454615735251262, 6.72771743377845975814782072264, 8.292510137016876784746702435083, 10.65453206642566383157798439612, 12.24183330898912093887117852077, 13.61658734895653415719031771991, 15.05353004836940420022726207348, 16.51612927015230562071161976443, 17.23913788753085667181037678850

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