Properties

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

Origins

Related objects

Downloads

Normalization:  

Dirichlet series

L(s)  = 1  − 24·2-s + 252·3-s − 1.47e3·4-s + 4.83e3·5-s − 6.04e3·6-s − 1.67e4·7-s + 8.44e4·8-s − 1.13e5·9-s − 1.15e5·10-s + 5.34e5·11-s − 3.70e5·12-s − 5.77e5·13-s + 4.01e5·14-s + 1.21e6·15-s + 9.87e5·16-s − 6.90e6·17-s + 2.72e6·18-s + 1.06e7·19-s − 7.10e6·20-s − 4.21e6·21-s − 1.28e7·22-s + 1.86e7·23-s + 2.12e7·24-s − 2.54e7·25-s + 1.38e7·26-s − 7.32e7·27-s + 2.46e7·28-s + ⋯
L(s)  = 1  − 0.530·2-s + 0.598·3-s − 0.718·4-s + 0.691·5-s − 0.317·6-s − 0.376·7-s + 0.911·8-s − 0.641·9-s − 0.366·10-s + 1.00·11-s − 0.430·12-s − 0.431·13-s + 0.199·14-s + 0.413·15-s + 0.235·16-s − 1.17·17-s + 0.340·18-s + 0.987·19-s − 0.496·20-s − 0.225·21-s − 0.530·22-s + 0.603·23-s + 0.545·24-s − 0.522·25-s + 0.228·26-s − 0.982·27-s + 0.270·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \,\Lambda(12-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1\)    =    \(1\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(11\)
character  :  $\chi_{1} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 1,\ (\ :11/2),\ 1)$
$L(6)$  $\approx$  $0.792122$
$L(\frac12)$  $\approx$  $0.792122$
$L(\frac{13}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, \(F_p\) is a polynomial of degree 2.
$p$$F_p$
good2 \( 1 + 3 p^{3} T + p^{11} T^{2} \)
3 \( 1 - 28 p^{2} T + p^{11} T^{2} \)
5 \( 1 - 966 p T + p^{11} T^{2} \)
7 \( 1 + 2392 p T + p^{11} T^{2} \)
11 \( 1 - 534612 T + p^{11} T^{2} \)
13 \( 1 + 577738 T + p^{11} T^{2} \)
17 \( 1 + 6905934 T + p^{11} T^{2} \)
19 \( 1 - 10661420 T + p^{11} T^{2} \)
23 \( 1 - 18643272 T + p^{11} T^{2} \)
29 \( 1 - 128406630 T + p^{11} T^{2} \)
31 \( 1 + 52843168 T + p^{11} T^{2} \)
37 \( 1 + 182213314 T + p^{11} T^{2} \)
41 \( 1 - 308120442 T + p^{11} T^{2} \)
43 \( 1 + 17125708 T + p^{11} T^{2} \)
47 \( 1 - 2687348496 T + p^{11} T^{2} \)
53 \( 1 + 1596055698 T + p^{11} T^{2} \)
59 \( 1 + 5189203740 T + p^{11} T^{2} \)
61 \( 1 - 6956478662 T + p^{11} T^{2} \)
67 \( 1 + 15481826884 T + p^{11} T^{2} \)
71 \( 1 - 9791485272 T + p^{11} T^{2} \)
73 \( 1 - 1463791322 T + p^{11} T^{2} \)
79 \( 1 - 38116845680 T + p^{11} T^{2} \)
83 \( 1 + 29335099668 T + p^{11} T^{2} \)
89 \( 1 + 24992917110 T + p^{11} T^{2} \)
97 \( 1 - 75013568546 T + p^{11} T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−32.77487538223120744183045567331, −31.17820949836025906449218889077, −28.83168262418687544502196191298, −26.80439115835040303257574923358, −25.27463654811236535674532419313, −22.33610363720986727568267445924, −19.65651314195496100012728175632, −17.44277697823447331355152513713, −13.90754986139213440644668132877, −9.222379399921102522243767192743, 9.222379399921102522243767192743, 13.90754986139213440644668132877, 17.44277697823447331355152513713, 19.65651314195496100012728175632, 22.33610363720986727568267445924, 25.27463654811236535674532419313, 26.80439115835040303257574923358, 28.83168262418687544502196191298, 31.17820949836025906449218889077, 32.77487538223120744183045567331

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