Properties

Degree 2
Conductor $ 2 \cdot 3^{3} \cdot 149 $
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  − 2-s + 4-s + 4.11·5-s − 3.78·7-s − 8-s − 4.11·10-s + 2.15·11-s + 5.55·13-s + 3.78·14-s + 16-s − 2.24·17-s + 2.44·19-s + 4.11·20-s − 2.15·22-s − 0.398·23-s + 11.8·25-s − 5.55·26-s − 3.78·28-s + 8.12·29-s + 0.810·31-s − 32-s + 2.24·34-s − 15.5·35-s + 8.17·37-s − 2.44·38-s − 4.11·40-s + 0.726·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + 1.83·5-s − 1.43·7-s − 0.353·8-s − 1.29·10-s + 0.649·11-s + 1.54·13-s + 1.01·14-s + 0.250·16-s − 0.545·17-s + 0.560·19-s + 0.919·20-s − 0.458·22-s − 0.0831·23-s + 2.37·25-s − 1.08·26-s − 0.715·28-s + 1.50·29-s + 0.145·31-s − 0.176·32-s + 0.385·34-s − 2.62·35-s + 1.34·37-s − 0.396·38-s − 0.649·40-s + 0.113·41-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8046\)    =    \(2 \cdot 3^{3} \cdot 149\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{8046} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 8046,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.215905226$
$L(\frac12)$  $\approx$  $2.215905226$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;149\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;149\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
3 \( 1 \)
149 \( 1 - T \)
good5 \( 1 - 4.11T + 5T^{2} \)
7 \( 1 + 3.78T + 7T^{2} \)
11 \( 1 - 2.15T + 11T^{2} \)
13 \( 1 - 5.55T + 13T^{2} \)
17 \( 1 + 2.24T + 17T^{2} \)
19 \( 1 - 2.44T + 19T^{2} \)
23 \( 1 + 0.398T + 23T^{2} \)
29 \( 1 - 8.12T + 29T^{2} \)
31 \( 1 - 0.810T + 31T^{2} \)
37 \( 1 - 8.17T + 37T^{2} \)
41 \( 1 - 0.726T + 41T^{2} \)
43 \( 1 + 3.96T + 43T^{2} \)
47 \( 1 - 7.08T + 47T^{2} \)
53 \( 1 + 9.53T + 53T^{2} \)
59 \( 1 + 11.4T + 59T^{2} \)
61 \( 1 - 5.54T + 61T^{2} \)
67 \( 1 - 3.37T + 67T^{2} \)
71 \( 1 - 8.38T + 71T^{2} \)
73 \( 1 + 5.85T + 73T^{2} \)
79 \( 1 + 10.4T + 79T^{2} \)
83 \( 1 + 1.42T + 83T^{2} \)
89 \( 1 - 15.6T + 89T^{2} \)
97 \( 1 + 10.0T + 97T^{2} \)
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

−7.961871384434964691945984262384, −6.78575936487506802213817793489, −6.39453672914108574124000846296, −6.15012034604870211688711757881, −5.35550083782714583210874043096, −4.20151038116311911100456919401, −3.17053157999225323626638204549, −2.63539162814471679983821388275, −1.58752221885432939926708736138, −0.879172602861975477086570577663, 0.879172602861975477086570577663, 1.58752221885432939926708736138, 2.63539162814471679983821388275, 3.17053157999225323626638204549, 4.20151038116311911100456919401, 5.35550083782714583210874043096, 6.15012034604870211688711757881, 6.39453672914108574124000846296, 6.78575936487506802213817793489, 7.961871384434964691945984262384

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