Properties

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

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 24.8·2-s + 81·3-s + 107.·4-s − 625·5-s − 2.01e3·6-s − 4.01e3·7-s + 1.00e4·8-s + 6.56e3·9-s + 1.55e4·10-s + 8.48e4·11-s + 8.68e3·12-s + 1.19e5·13-s + 9.97e4·14-s − 5.06e4·15-s − 3.05e5·16-s + 1.16e5·17-s − 1.63e5·18-s − 2.34e5·19-s − 6.70e4·20-s − 3.24e5·21-s − 2.11e6·22-s + 2.34e6·23-s + 8.15e5·24-s + 3.90e5·25-s − 2.97e6·26-s + 5.31e5·27-s − 4.29e5·28-s + ⋯
L(s)  = 1  − 1.09·2-s + 0.577·3-s + 0.209·4-s − 0.447·5-s − 0.634·6-s − 0.631·7-s + 0.869·8-s + 0.333·9-s + 0.491·10-s + 1.74·11-s + 0.120·12-s + 1.15·13-s + 0.694·14-s − 0.258·15-s − 1.16·16-s + 0.339·17-s − 0.366·18-s − 0.413·19-s − 0.0936·20-s − 0.364·21-s − 1.92·22-s + 1.74·23-s + 0.501·24-s + 0.200·25-s − 1.27·26-s + 0.192·27-s − 0.132·28-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  =  \((2,\ 15,\ (\ :9/2),\ 1)\)
\(L(5)\)  \(\approx\)  \(1.04681\)
\(L(\frac12)\)  \(\approx\)  \(1.04681\)
\(L(\frac{11}{2})\)   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 \( 1 + 24.8T + 512T^{2} \)
7 \( 1 + 4.01e3T + 4.03e7T^{2} \)
11 \( 1 - 8.48e4T + 2.35e9T^{2} \)
13 \( 1 - 1.19e5T + 1.06e10T^{2} \)
17 \( 1 - 1.16e5T + 1.18e11T^{2} \)
19 \( 1 + 2.34e5T + 3.22e11T^{2} \)
23 \( 1 - 2.34e6T + 1.80e12T^{2} \)
29 \( 1 + 4.64e5T + 1.45e13T^{2} \)
31 \( 1 + 5.11e6T + 2.64e13T^{2} \)
37 \( 1 - 8.69e6T + 1.29e14T^{2} \)
41 \( 1 + 9.05e6T + 3.27e14T^{2} \)
43 \( 1 - 8.63e6T + 5.02e14T^{2} \)
47 \( 1 - 3.31e7T + 1.11e15T^{2} \)
53 \( 1 + 6.41e7T + 3.29e15T^{2} \)
59 \( 1 - 1.49e8T + 8.66e15T^{2} \)
61 \( 1 - 1.54e8T + 1.16e16T^{2} \)
67 \( 1 - 2.72e8T + 2.72e16T^{2} \)
71 \( 1 + 3.56e8T + 4.58e16T^{2} \)
73 \( 1 - 2.06e8T + 5.88e16T^{2} \)
79 \( 1 + 4.04e8T + 1.19e17T^{2} \)
83 \( 1 + 5.17e6T + 1.86e17T^{2} \)
89 \( 1 - 4.32e8T + 3.50e17T^{2} \)
97 \( 1 + 1.32e9T + 7.60e17T^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−17.21074291978532356264856715795, −16.15918161978580588306669409584, −14.56252578855415426272620302290, −13.06114903251926765807882633844, −11.14125365410158502085590269814, −9.454364224001635162685792574033, −8.584576281800275813544804282475, −6.89779808833438444439512651130, −3.81141853109217920931067793814, −1.10701398784709352907941481180, 1.10701398784709352907941481180, 3.81141853109217920931067793814, 6.89779808833438444439512651130, 8.584576281800275813544804282475, 9.454364224001635162685792574033, 11.14125365410158502085590269814, 13.06114903251926765807882633844, 14.56252578855415426272620302290, 16.15918161978580588306669409584, 17.21074291978532356264856715795

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