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Properties

Label 2-15-15.14-c8-0-5
Degree $2$
Conductor $15$
Sign $0.337 - 0.941i$
Analytic cond. $6.11067$
Root an. cond. $2.47197$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 17.5·2-s + (74.7 + 31.2i)3-s + 53.2·4-s + (−31.8 − 624. i)5-s + (−1.31e3 − 549. i)6-s + 3.42e3i·7-s + 3.56e3·8-s + (4.61e3 + 4.66e3i)9-s + (560. + 1.09e4i)10-s + 1.57e4i·11-s + (3.97e3 + 1.66e3i)12-s + 2.70e4i·13-s − 6.02e4i·14-s + (1.71e4 − 4.76e4i)15-s − 7.63e4·16-s − 1.42e4·17-s + ⋯
L(s)  = 1  − 1.09·2-s + (0.922 + 0.385i)3-s + 0.207·4-s + (−0.0509 − 0.998i)5-s + (−1.01 − 0.423i)6-s + 1.42i·7-s + 0.870·8-s + (0.702 + 0.711i)9-s + (0.0560 + 1.09i)10-s + 1.07i·11-s + (0.191 + 0.0801i)12-s + 0.947i·13-s − 1.56i·14-s + (0.337 − 0.941i)15-s − 1.16·16-s − 0.170·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.337 - 0.941i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.337 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $0.337 - 0.941i$
Analytic conductor: \(6.11067\)
Root analytic conductor: \(2.47197\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{15} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :4),\ 0.337 - 0.941i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.886051 + 0.623255i\)
\(L(\frac12)\) \(\approx\) \(0.886051 + 0.623255i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-74.7 - 31.2i)T \)
5 \( 1 + (31.8 + 624. i)T \)
good2 \( 1 + 17.5T + 256T^{2} \)
7 \( 1 - 3.42e3iT - 5.76e6T^{2} \)
11 \( 1 - 1.57e4iT - 2.14e8T^{2} \)
13 \( 1 - 2.70e4iT - 8.15e8T^{2} \)
17 \( 1 + 1.42e4T + 6.97e9T^{2} \)
19 \( 1 - 2.10e5T + 1.69e10T^{2} \)
23 \( 1 + 6.02e4T + 7.83e10T^{2} \)
29 \( 1 + 7.58e5iT - 5.00e11T^{2} \)
31 \( 1 - 3.33e5T + 8.52e11T^{2} \)
37 \( 1 - 6.83e5iT - 3.51e12T^{2} \)
41 \( 1 + 1.23e6iT - 7.98e12T^{2} \)
43 \( 1 - 1.15e6iT - 1.16e13T^{2} \)
47 \( 1 + 1.52e6T + 2.38e13T^{2} \)
53 \( 1 + 4.73e6T + 6.22e13T^{2} \)
59 \( 1 - 1.30e7iT - 1.46e14T^{2} \)
61 \( 1 + 6.62e6T + 1.91e14T^{2} \)
67 \( 1 + 2.47e7iT - 4.06e14T^{2} \)
71 \( 1 + 1.07e7iT - 6.45e14T^{2} \)
73 \( 1 + 3.48e7iT - 8.06e14T^{2} \)
79 \( 1 - 5.01e7T + 1.51e15T^{2} \)
83 \( 1 - 4.50e7T + 2.25e15T^{2} \)
89 \( 1 + 7.30e7iT - 3.93e15T^{2} \)
97 \( 1 - 1.45e8iT - 7.83e15T^{2} \)
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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

−17.89901536355079462762676573275, −16.37318228928669548684759865746, −15.37180623280658018225313518108, −13.65222498826019672336681832345, −12.03313929943111565806038881479, −9.640829350486948604073039951430, −9.095569040270967642928137409087, −7.84807450125697887569805623090, −4.72818753667012009286496519660, −1.86899966701895068496293685381, 0.886333222263609467557592861088, 3.39986834428288870258730783434, 7.13106292759717051892759432925, 8.082167845273079201919403140755, 9.804094116014769503645563319538, 10.88773603285723660287293290118, 13.48077125845241069483205052738, 14.21920097039191217536495752033, 16.05214943385436893972193128280, 17.63060310911507872398360078864

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