Properties

Degree 2
Conductor $ 2^{3} \cdot 3797 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 3

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 4·5-s − 2·7-s + 9-s − 6·11-s − 13-s + 8·15-s − 8·17-s − 7·19-s + 4·21-s − 9·23-s + 11·25-s + 4·27-s − 6·29-s − 6·31-s + 12·33-s + 8·35-s − 4·37-s + 2·39-s − 2·41-s + 5·43-s − 4·45-s − 3·49-s + 16·51-s − 6·53-s + 24·55-s + 14·57-s + ⋯
L(s)  = 1  − 1.15·3-s − 1.78·5-s − 0.755·7-s + 1/3·9-s − 1.80·11-s − 0.277·13-s + 2.06·15-s − 1.94·17-s − 1.60·19-s + 0.872·21-s − 1.87·23-s + 11/5·25-s + 0.769·27-s − 1.11·29-s − 1.07·31-s + 2.08·33-s + 1.35·35-s − 0.657·37-s + 0.320·39-s − 0.312·41-s + 0.762·43-s − 0.596·45-s − 3/7·49-s + 2.24·51-s − 0.824·53-s + 3.23·55-s + 1.85·57-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 30376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 30376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(30376\)    =    \(2^{3} \cdot 3797\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{30376} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  3
Selberg data  =  $(2,\ 30376,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3797\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3797\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
3797 \( 1 + T \)
good3 \( 1 + 2 T + p T^{2} \)
5 \( 1 + 4 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + 8 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 + 9 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 5 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 + T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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

−15.92595014189507, −15.64234861513119, −15.08083110328672, −14.55843154946923, −13.58199547138613, −12.97073328959031, −12.67243219943437, −12.23389790931235, −11.59378100753876, −11.04161857036807, −10.76860725672937, −10.37686057515731, −9.507890997338932, −8.595562855046562, −8.384022462641684, −7.573758829419918, −7.201200827403869, −6.487796721061515, −5.979993841803799, −5.310543461569214, −4.476114756748991, −4.303826073777747, −3.465613178018053, −2.663208327258735, −1.938045659864408, 0, 0, 0, 1.938045659864408, 2.663208327258735, 3.465613178018053, 4.303826073777747, 4.476114756748991, 5.310543461569214, 5.979993841803799, 6.487796721061515, 7.201200827403869, 7.573758829419918, 8.384022462641684, 8.595562855046562, 9.507890997338932, 10.37686057515731, 10.76860725672937, 11.04161857036807, 11.59378100753876, 12.23389790931235, 12.67243219943437, 12.97073328959031, 13.58199547138613, 14.55843154946923, 15.08083110328672, 15.64234861513119, 15.92595014189507

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