Properties

Degree 2
Conductor $ 3 \cdot 5 $
Sign $1$
Motivic weight 9
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 43.8·2-s + 81·3-s + 1.41e3·4-s − 625·5-s + 3.55e3·6-s − 7.86e3·7-s + 3.95e4·8-s + 6.56e3·9-s − 2.74e4·10-s − 4.93e4·11-s + 1.14e5·12-s + 2.42e4·13-s − 3.44e5·14-s − 5.06e4·15-s + 1.01e6·16-s + 2.68e5·17-s + 2.87e5·18-s − 1.68e5·19-s − 8.83e5·20-s − 6.36e5·21-s − 2.16e6·22-s − 2.12e6·23-s + 3.20e6·24-s + 3.90e5·25-s + 1.06e6·26-s + 5.31e5·27-s − 1.11e7·28-s + ⋯
L(s)  = 1  + 1.93·2-s + 0.577·3-s + 2.76·4-s − 0.447·5-s + 1.11·6-s − 1.23·7-s + 3.41·8-s + 0.333·9-s − 0.867·10-s − 1.01·11-s + 1.59·12-s + 0.235·13-s − 2.40·14-s − 0.258·15-s + 3.86·16-s + 0.778·17-s + 0.646·18-s − 0.296·19-s − 1.23·20-s − 0.714·21-s − 1.97·22-s − 1.58·23-s + 1.97·24-s + 0.200·25-s + 0.456·26-s + 0.192·27-s − 3.41·28-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(15\)    =    \(3 \cdot 5\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(9\)
character  :  $\chi_{15} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 15,\ (\ :9/2),\ 1)\)
\(L(5)\)  \(\approx\)  \(4.99210\)
\(L(\frac12)\)  \(\approx\)  \(4.99210\)
\(L(\frac{11}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{3,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 - 81T \)
5 \( 1 + 625T \)
good2 \( 1 - 43.8T + 512T^{2} \)
7 \( 1 + 7.86e3T + 4.03e7T^{2} \)
11 \( 1 + 4.93e4T + 2.35e9T^{2} \)
13 \( 1 - 2.42e4T + 1.06e10T^{2} \)
17 \( 1 - 2.68e5T + 1.18e11T^{2} \)
19 \( 1 + 1.68e5T + 3.22e11T^{2} \)
23 \( 1 + 2.12e6T + 1.80e12T^{2} \)
29 \( 1 - 3.89e5T + 1.45e13T^{2} \)
31 \( 1 - 9.05e4T + 2.64e13T^{2} \)
37 \( 1 + 3.31e6T + 1.29e14T^{2} \)
41 \( 1 - 2.32e7T + 3.27e14T^{2} \)
43 \( 1 - 1.91e7T + 5.02e14T^{2} \)
47 \( 1 - 6.28e7T + 1.11e15T^{2} \)
53 \( 1 + 1.80e5T + 3.29e15T^{2} \)
59 \( 1 - 3.84e7T + 8.66e15T^{2} \)
61 \( 1 + 5.53e5T + 1.16e16T^{2} \)
67 \( 1 + 2.39e8T + 2.72e16T^{2} \)
71 \( 1 - 1.28e8T + 4.58e16T^{2} \)
73 \( 1 + 2.39e8T + 5.88e16T^{2} \)
79 \( 1 + 5.28e8T + 1.19e17T^{2} \)
83 \( 1 - 2.12e8T + 1.86e17T^{2} \)
89 \( 1 + 2.07e8T + 3.50e17T^{2} \)
97 \( 1 - 1.70e9T + 7.60e17T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−16.19540715748169142862627741942, −15.61423861807197255757040028166, −14.23478208490347068356780115683, −13.11378504823525760733677885669, −12.20877336711187005618390922722, −10.40107408811533908524260023946, −7.54293670634805485007182826298, −5.91994363575284975170327005585, −3.96891381680544607615331011090, −2.70407076350904942917171515428, 2.70407076350904942917171515428, 3.96891381680544607615331011090, 5.91994363575284975170327005585, 7.54293670634805485007182826298, 10.40107408811533908524260023946, 12.20877336711187005618390922722, 13.11378504823525760733677885669, 14.23478208490347068356780115683, 15.61423861807197255757040028166, 16.19540715748169142862627741942

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