Properties

Degree 2
Conductor $ 3 \cdot 5 $
Sign $1$
Motivic weight 7
Primitive yes
Self-dual yes
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 15.7·2-s + 27·3-s + 120.·4-s + 125·5-s + 425.·6-s − 34.4·7-s − 121.·8-s + 729·9-s + 1.96e3·10-s − 3.96e3·11-s + 3.24e3·12-s + 5.60e3·13-s − 542.·14-s + 3.37e3·15-s − 1.73e4·16-s − 1.99e4·17-s + 1.14e4·18-s − 4.99e4·19-s + 1.50e4·20-s − 929.·21-s − 6.24e4·22-s + 1.09e5·23-s − 3.27e3·24-s + 1.56e4·25-s + 8.83e4·26-s + 1.96e4·27-s − 4.14e3·28-s + ⋯
L(s)  = 1  + 1.39·2-s + 0.577·3-s + 0.939·4-s + 0.447·5-s + 0.804·6-s − 0.0379·7-s − 0.0837·8-s + 0.333·9-s + 0.622·10-s − 0.897·11-s + 0.542·12-s + 0.707·13-s − 0.0528·14-s + 0.258·15-s − 1.05·16-s − 0.982·17-s + 0.464·18-s − 1.67·19-s + 0.420·20-s − 0.0219·21-s − 1.25·22-s + 1.88·23-s − 0.0483·24-s + 0.199·25-s + 0.985·26-s + 0.192·27-s − 0.0356·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(15\)    =    \(3 \cdot 5\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(7\)
character  :  $\chi_{15} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 15,\ (\ :7/2),\ 1)$
$L(4)$  $\approx$  $3.32257$
$L(\frac12)$  $\approx$  $3.32257$
$L(\frac{9}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;5\}$, \(F_p\) is a polynomial of degree 2. If $p \in \{3,\;5\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 - 27T \)
5 \( 1 - 125T \)
good2 \( 1 - 15.7T + 128T^{2} \)
7 \( 1 + 34.4T + 8.23e5T^{2} \)
11 \( 1 + 3.96e3T + 1.94e7T^{2} \)
13 \( 1 - 5.60e3T + 6.27e7T^{2} \)
17 \( 1 + 1.99e4T + 4.10e8T^{2} \)
19 \( 1 + 4.99e4T + 8.93e8T^{2} \)
23 \( 1 - 1.09e5T + 3.40e9T^{2} \)
29 \( 1 - 1.92e5T + 1.72e10T^{2} \)
31 \( 1 - 1.25e5T + 2.75e10T^{2} \)
37 \( 1 + 7.43e4T + 9.49e10T^{2} \)
41 \( 1 - 5.77e5T + 1.94e11T^{2} \)
43 \( 1 - 2.64e5T + 2.71e11T^{2} \)
47 \( 1 + 3.06e5T + 5.06e11T^{2} \)
53 \( 1 + 4.46e5T + 1.17e12T^{2} \)
59 \( 1 - 1.97e6T + 2.48e12T^{2} \)
61 \( 1 + 1.27e6T + 3.14e12T^{2} \)
67 \( 1 + 4.12e6T + 6.06e12T^{2} \)
71 \( 1 + 2.81e6T + 9.09e12T^{2} \)
73 \( 1 - 4.01e6T + 1.10e13T^{2} \)
79 \( 1 + 1.32e6T + 1.92e13T^{2} \)
83 \( 1 + 1.91e6T + 2.71e13T^{2} \)
89 \( 1 - 8.00e6T + 4.42e13T^{2} \)
97 \( 1 + 3.89e6T + 8.07e13T^{2} \)
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

−17.74919663695131842440082232171, −15.79105659830712106912656777392, −14.78061773736410346579605067880, −13.47892113249999878498918010377, −12.80851836828137777524822912202, −10.83335945020528956171161022592, −8.760884674874884043127220695129, −6.44363249489805156692362086648, −4.62351224268796633110936965482, −2.70596641029739944237853956738, 2.70596641029739944237853956738, 4.62351224268796633110936965482, 6.44363249489805156692362086648, 8.760884674874884043127220695129, 10.83335945020528956171161022592, 12.80851836828137777524822912202, 13.47892113249999878498918010377, 14.78061773736410346579605067880, 15.79105659830712106912656777392, 17.74919663695131842440082232171

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