Properties

Degree 2
Conductor $ 5 \cdot 11 \cdot 73 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 0.199·2-s − 3.23·3-s − 1.96·4-s + 5-s + 0.645·6-s − 5.08·7-s + 0.790·8-s + 7.45·9-s − 0.199·10-s + 11-s + 6.33·12-s + 1.81·13-s + 1.01·14-s − 3.23·15-s + 3.76·16-s + 2.44·17-s − 1.48·18-s − 3.81·19-s − 1.96·20-s + 16.4·21-s − 0.199·22-s + 1.59·23-s − 2.55·24-s + 25-s − 0.361·26-s − 14.4·27-s + 9.97·28-s + ⋯
L(s)  = 1  − 0.141·2-s − 1.86·3-s − 0.980·4-s + 0.447·5-s + 0.263·6-s − 1.92·7-s + 0.279·8-s + 2.48·9-s − 0.0630·10-s + 0.301·11-s + 1.82·12-s + 0.503·13-s + 0.271·14-s − 0.834·15-s + 0.940·16-s + 0.593·17-s − 0.350·18-s − 0.874·19-s − 0.438·20-s + 3.59·21-s − 0.0425·22-s + 0.333·23-s − 0.521·24-s + 0.200·25-s − 0.0709·26-s − 2.77·27-s + 1.88·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4015\)    =    \(5 \cdot 11 \cdot 73\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4015} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 4015,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.3030561384$
$L(\frac12)$  $\approx$  $0.3030561384$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{5,\;11,\;73\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{5,\;11,\;73\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad5 \( 1 - T \)
11 \( 1 - T \)
73 \( 1 - T \)
good2 \( 1 + 0.199T + 2T^{2} \)
3 \( 1 + 3.23T + 3T^{2} \)
7 \( 1 + 5.08T + 7T^{2} \)
13 \( 1 - 1.81T + 13T^{2} \)
17 \( 1 - 2.44T + 17T^{2} \)
19 \( 1 + 3.81T + 19T^{2} \)
23 \( 1 - 1.59T + 23T^{2} \)
29 \( 1 + 0.667T + 29T^{2} \)
31 \( 1 + 0.936T + 31T^{2} \)
37 \( 1 + 9.86T + 37T^{2} \)
41 \( 1 - 1.75T + 41T^{2} \)
43 \( 1 + 1.98T + 43T^{2} \)
47 \( 1 - 8.88T + 47T^{2} \)
53 \( 1 + 7.64T + 53T^{2} \)
59 \( 1 + 11.6T + 59T^{2} \)
61 \( 1 + 4.97T + 61T^{2} \)
67 \( 1 + 13.1T + 67T^{2} \)
71 \( 1 - 7.86T + 71T^{2} \)
79 \( 1 + 3.69T + 79T^{2} \)
83 \( 1 + 15.6T + 83T^{2} \)
89 \( 1 - 1.50T + 89T^{2} \)
97 \( 1 + 2.22T + 97T^{2} \)
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

−8.727428402036933537761864303574, −7.39791024633333440968324741869, −6.74222321087496118800926440606, −6.00907758237977674879760248983, −5.76349066970330007407948139650, −4.81050355853050032541605505138, −4.02156418218475670499868475413, −3.23360500208082317986724390989, −1.45188416573991716864206229714, −0.38085981519290834752066702583, 0.38085981519290834752066702583, 1.45188416573991716864206229714, 3.23360500208082317986724390989, 4.02156418218475670499868475413, 4.81050355853050032541605505138, 5.76349066970330007407948139650, 6.00907758237977674879760248983, 6.74222321087496118800926440606, 7.39791024633333440968324741869, 8.727428402036933537761864303574

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