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Properties

Label 2-15-15.14-c6-0-0
Degree $2$
Conductor $15$
Sign $-0.978 - 0.204i$
Analytic cond. $3.45081$
Root an. cond. $1.85763$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.12·2-s + (−19.8 + 18.2i)3-s − 59.4·4-s + (−72.7 − 101. i)5-s + (−42.2 + 38.8i)6-s + 435. i·7-s − 262.·8-s + (60.6 − 726. i)9-s + (−154. − 216. i)10-s + 1.65e3i·11-s + (1.18e3 − 1.08e3i)12-s − 1.64e3i·13-s + 926. i·14-s + (3.30e3 + 689. i)15-s + 3.24e3·16-s − 2.05e3·17-s + ⋯
L(s)  = 1  + 0.265·2-s + (−0.735 + 0.677i)3-s − 0.929·4-s + (−0.582 − 0.813i)5-s + (−0.195 + 0.180i)6-s + 1.26i·7-s − 0.512·8-s + (0.0831 − 0.996i)9-s + (−0.154 − 0.216i)10-s + 1.24i·11-s + (0.683 − 0.629i)12-s − 0.747i·13-s + 0.337i·14-s + (0.978 + 0.204i)15-s + 0.792·16-s − 0.417·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 - 0.204i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.978 - 0.204i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $-0.978 - 0.204i$
Analytic conductor: \(3.45081\)
Root analytic conductor: \(1.85763\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{15} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :3),\ -0.978 - 0.204i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.0322642 + 0.312575i\)
\(L(\frac12)\) \(\approx\) \(0.0322642 + 0.312575i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (19.8 - 18.2i)T \)
5 \( 1 + (72.7 + 101. i)T \)
good2 \( 1 - 2.12T + 64T^{2} \)
7 \( 1 - 435. iT - 1.17e5T^{2} \)
11 \( 1 - 1.65e3iT - 1.77e6T^{2} \)
13 \( 1 + 1.64e3iT - 4.82e6T^{2} \)
17 \( 1 + 2.05e3T + 2.41e7T^{2} \)
19 \( 1 + 3.63e3T + 4.70e7T^{2} \)
23 \( 1 + 2.12e4T + 1.48e8T^{2} \)
29 \( 1 - 1.14e4iT - 5.94e8T^{2} \)
31 \( 1 - 1.34e4T + 8.87e8T^{2} \)
37 \( 1 + 1.77e3iT - 2.56e9T^{2} \)
41 \( 1 - 2.29e3iT - 4.75e9T^{2} \)
43 \( 1 + 1.44e4iT - 6.32e9T^{2} \)
47 \( 1 + 4.96e4T + 1.07e10T^{2} \)
53 \( 1 - 8.17e4T + 2.21e10T^{2} \)
59 \( 1 - 2.00e5iT - 4.21e10T^{2} \)
61 \( 1 - 6.37e4T + 5.15e10T^{2} \)
67 \( 1 + 3.51e5iT - 9.04e10T^{2} \)
71 \( 1 - 4.04e5iT - 1.28e11T^{2} \)
73 \( 1 - 6.66e5iT - 1.51e11T^{2} \)
79 \( 1 + 7.58e5T + 2.43e11T^{2} \)
83 \( 1 - 2.61e5T + 3.26e11T^{2} \)
89 \( 1 + 3.16e5iT - 4.96e11T^{2} \)
97 \( 1 - 3.20e5iT - 8.32e11T^{2} \)
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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

−18.29885913604611387213621120307, −17.40263266174661977727616398330, −15.79197649346041549519522996672, −14.94399335404258781904021864039, −12.70924894734526228316602200055, −11.99545411477445886512846122146, −9.840231454516263936477964189644, −8.564441561128573738316790986821, −5.55527682887432286834119884247, −4.32448210734120036902524704134, 0.23268793191604961560974224557, 4.09508038044328891778579596216, 6.36795943408830265416563294029, 8.011763349663693446892611499749, 10.45741544851985405704618206477, 11.72309640880427805714239521282, 13.44934918157026481534854210183, 14.18423546892082179586858020893, 16.30339162393016893026330471711, 17.54044857104208073514531611640

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