Properties

Degree 2
Conductor $ 37 \cdot 109 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2.50·2-s + 1.55·3-s + 4.29·4-s − 3.66·5-s − 3.90·6-s + 4.73·7-s − 5.76·8-s − 0.575·9-s + 9.19·10-s + 5.22·11-s + 6.69·12-s − 3.90·13-s − 11.8·14-s − 5.70·15-s + 5.87·16-s − 3.68·17-s + 1.44·18-s − 5.02·19-s − 15.7·20-s + 7.37·21-s − 13.1·22-s − 0.531·23-s − 8.97·24-s + 8.41·25-s + 9.78·26-s − 5.56·27-s + 20.3·28-s + ⋯
L(s)  = 1  − 1.77·2-s + 0.899·3-s + 2.14·4-s − 1.63·5-s − 1.59·6-s + 1.79·7-s − 2.03·8-s − 0.191·9-s + 2.90·10-s + 1.57·11-s + 1.93·12-s − 1.08·13-s − 3.17·14-s − 1.47·15-s + 1.46·16-s − 0.893·17-s + 0.340·18-s − 1.15·19-s − 3.51·20-s + 1.61·21-s − 2.79·22-s − 0.110·23-s − 1.83·24-s + 1.68·25-s + 1.91·26-s − 1.07·27-s + 3.84·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4033\)    =    \(37 \cdot 109\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{4033} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 4033,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{37,\;109\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{37,\;109\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad37 \( 1 + T \)
109 \( 1 + T \)
good2 \( 1 + 2.50T + 2T^{2} \)
3 \( 1 - 1.55T + 3T^{2} \)
5 \( 1 + 3.66T + 5T^{2} \)
7 \( 1 - 4.73T + 7T^{2} \)
11 \( 1 - 5.22T + 11T^{2} \)
13 \( 1 + 3.90T + 13T^{2} \)
17 \( 1 + 3.68T + 17T^{2} \)
19 \( 1 + 5.02T + 19T^{2} \)
23 \( 1 + 0.531T + 23T^{2} \)
29 \( 1 - 7.93T + 29T^{2} \)
31 \( 1 - 9.09T + 31T^{2} \)
41 \( 1 + 9.30T + 41T^{2} \)
43 \( 1 + 1.44T + 43T^{2} \)
47 \( 1 + 4.67T + 47T^{2} \)
53 \( 1 + 9.23T + 53T^{2} \)
59 \( 1 + 13.6T + 59T^{2} \)
61 \( 1 - 7.43T + 61T^{2} \)
67 \( 1 - 6.71T + 67T^{2} \)
71 \( 1 + 3.82T + 71T^{2} \)
73 \( 1 + 14.4T + 73T^{2} \)
79 \( 1 - 3.11T + 79T^{2} \)
83 \( 1 + 6.20T + 83T^{2} \)
89 \( 1 + 2.99T + 89T^{2} \)
97 \( 1 - 16.5T + 97T^{2} \)
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

−8.198098559325052850330860535956, −7.88388824451330446602508982519, −6.99766525430574856753953884969, −6.49214129323379457974875302232, −4.67451197408775782796593021715, −4.31194248628931202416587303320, −3.07668890495502892368872042957, −2.14820628053939124891354447983, −1.30171591971930536460937028209, 0, 1.30171591971930536460937028209, 2.14820628053939124891354447983, 3.07668890495502892368872042957, 4.31194248628931202416587303320, 4.67451197408775782796593021715, 6.49214129323379457974875302232, 6.99766525430574856753953884969, 7.88388824451330446602508982519, 8.198098559325052850330860535956

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