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Properties

Label 2-15-5.4-c9-0-5
Degree $2$
Conductor $15$
Sign $-0.157 + 0.987i$
Analytic cond. $7.72553$
Root an. cond. $2.77948$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 14.3i·2-s − 81i·3-s + 306.·4-s + (1.38e3 + 220. i)5-s − 1.16e3·6-s − 2.87e3i·7-s − 1.17e4i·8-s − 6.56e3·9-s + (3.16e3 − 1.97e4i)10-s − 2.32e4·11-s − 2.48e4i·12-s − 1.12e5i·13-s − 4.12e4·14-s + (1.78e4 − 1.11e5i)15-s − 1.13e4·16-s + 1.15e5i·17-s + ⋯
L(s)  = 1  − 0.633i·2-s − 0.577i·3-s + 0.598·4-s + (0.987 + 0.157i)5-s − 0.365·6-s − 0.453i·7-s − 1.01i·8-s − 0.333·9-s + (0.100 − 0.625i)10-s − 0.479·11-s − 0.345i·12-s − 1.09i·13-s − 0.287·14-s + (0.0911 − 0.570i)15-s − 0.0432·16-s + 0.336i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.157 + 0.987i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.157 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $-0.157 + 0.987i$
Analytic conductor: \(7.72553\)
Root analytic conductor: \(2.77948\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{15} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :9/2),\ -0.157 + 0.987i)\)

Particular Values

\(L(5)\) \(\approx\) \(1.39950 - 1.64112i\)
\(L(\frac12)\) \(\approx\) \(1.39950 - 1.64112i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 81iT \)
5 \( 1 + (-1.38e3 - 220. i)T \)
good2 \( 1 + 14.3iT - 512T^{2} \)
7 \( 1 + 2.87e3iT - 4.03e7T^{2} \)
11 \( 1 + 2.32e4T + 2.35e9T^{2} \)
13 \( 1 + 1.12e5iT - 1.06e10T^{2} \)
17 \( 1 - 1.15e5iT - 1.18e11T^{2} \)
19 \( 1 - 2.13e5T + 3.22e11T^{2} \)
23 \( 1 - 1.83e6iT - 1.80e12T^{2} \)
29 \( 1 + 3.67e6T + 1.45e13T^{2} \)
31 \( 1 - 8.85e6T + 2.64e13T^{2} \)
37 \( 1 + 9.17e6iT - 1.29e14T^{2} \)
41 \( 1 + 1.17e7T + 3.27e14T^{2} \)
43 \( 1 - 3.93e7iT - 5.02e14T^{2} \)
47 \( 1 - 3.26e7iT - 1.11e15T^{2} \)
53 \( 1 - 1.04e8iT - 3.29e15T^{2} \)
59 \( 1 - 1.02e8T + 8.66e15T^{2} \)
61 \( 1 + 1.65e8T + 1.16e16T^{2} \)
67 \( 1 + 1.76e8iT - 2.72e16T^{2} \)
71 \( 1 - 1.30e8T + 4.58e16T^{2} \)
73 \( 1 - 2.35e8iT - 5.88e16T^{2} \)
79 \( 1 + 1.56e8T + 1.19e17T^{2} \)
83 \( 1 - 3.38e7iT - 1.86e17T^{2} \)
89 \( 1 + 4.86e8T + 3.50e17T^{2} \)
97 \( 1 + 1.40e9iT - 7.60e17T^{2} \)
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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

−17.08897198667512279287806074827, −15.41401934036617589561742386352, −13.66395588685887816928927935115, −12.65199288226036189847937932192, −11.03515106935981839800070895643, −9.874620620656455339487418342591, −7.53064209529914651618619406866, −5.91832183949341266972285573683, −2.87852774425775009653577283932, −1.27286022515273165208726702548, 2.32569165219823342722075518409, 5.19400243384815031306730447756, 6.60750871499045465129459240750, 8.678165017215474271147753138645, 10.23009910685017842971714079552, 11.82105193048389862704174629362, 13.79108563576215519347444794762, 15.01959570062942589280964531715, 16.24684111688042782503396097971, 17.12007760453289626740830146824

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