Properties

Degree 2
Conductor 5077
Sign $-1$
Self-dual yes
Motivic weight 1

Origins

Related objects

Downloads

Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 3·3-s + 2·4-s − 4·5-s + 6·6-s − 4·7-s + 6·9-s + 8·10-s − 6·11-s − 6·12-s − 4·13-s + 8·14-s + 12·15-s − 4·16-s − 4·17-s − 12·18-s − 7·19-s − 8·20-s + 12·21-s + 12·22-s − 6·23-s + 11·25-s + 8·26-s − 9·27-s − 8·28-s − 6·29-s − 24·30-s + ⋯
L(s)  = 1  − 1.414·2-s − 1.732·3-s + 4-s − 1.788·5-s + 2.449·6-s − 1.511·7-s + 2·9-s + 2.529·10-s − 1.809·11-s − 1.732·12-s − 1.109·13-s + 2.138·14-s + 3.098·15-s − 16-s − 0.970·17-s − 2.828·18-s − 1.605·19-s − 1.788·20-s + 2.618·21-s + 2.558·22-s − 1.251·23-s + 2.2·25-s + 1.568·26-s − 1.732·27-s − 1.511·28-s − 1.114·29-s − 4.381·30-s + ⋯

Functional equation

\[\begin{align} \Lambda(s)=\mathstrut & 5077 ^{s/2} \Gamma_{\C}(s) \cdot L(s)\cr =\mathstrut & - \Lambda(2-s) \end{align} \]
\[\begin{align} \Lambda(s)=\mathstrut & 5077 ^{s/2} \Gamma_{\C}(s+0.5) \cdot L(s)\cr =\mathstrut & - \Lambda(1-s) \end{align} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(5077\)
\( \varepsilon \)  =  $-1$
weight  =  1
character  :  $\chi_{5077} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  3
Selberg data  =  $(2,\ 5077,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 5077$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p = 5077$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad5077$1+T$
good2$1+2T+2T^{2}$
3$1+3T+3T^{2}$
5$1+4T+5T^{2}$
7$1+4T+7T^{2}$
11$1+6T+11T^{2}$
13$1+4T+13T^{2}$
17$1+4T+17T^{2}$
19$1+7T+19T^{2}$
23$1+6T+23T^{2}$
29$1+6T+29T^{2}$
31$1+2T+31T^{2}$
37$1+37T^{2}$
41$1+41T^{2}$
43$1+8T+43T^{2}$
47$1+9T+47T^{2}$
53$1+9T+53T^{2}$
59$1+11T+59T^{2}$
61$1+2T+61T^{2}$
67$1+12T+67T^{2}$
71$1+8T+71T^{2}$
73$1+14T+73T^{2}$
79$1-9T+79T^{2}$
83$1+2T+83T^{2}$
89$1-11T+89T^{2}$
97$1-6T+97T^{2}$
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\[\begin{equation} L(s) = \prod_{p\ \mathrm{bad}} (1- a(p) p^{-s})^{-1} \prod_{p\ \mathrm{good}} (1- a(p) p^{-s} + p^{-2s})^{-1} \end{equation}\]

Imaginary part of the first few zeros on the critical line

−18.55973951718974, −18.07306090799613, −17.41152136149437, −16.64312940081154, −16.44048490106365, −15.89479293723709, −15.51534707653608, −14.94156032954847, −13.15163660315273, −12.97272258207286, −12.28722890382493, −11.68694809088531, −11.03344123514270, −10.49585360108396, −10.20346324266066, −9.382178911171940, −8.476801942623500, −7.706794648113253, −7.342814979539648, −6.622504613407707, −6.011922752986395, −4.754431515963406, −4.470551513310098, −3.262443555978757, −2.052472858479940, 0, 0, 0, 2.052472858479940, 3.262443555978757, 4.470551513310098, 4.754431515963406, 6.011922752986395, 6.622504613407707, 7.342814979539648, 7.706794648113253, 8.476801942623500, 9.382178911171940, 10.20346324266066, 10.49585360108396, 11.03344123514270, 11.68694809088531, 12.28722890382493, 12.97272258207286, 13.15163660315273, 14.94156032954847, 15.51534707653608, 15.89479293723709, 16.44048490106365, 16.64312940081154, 17.41152136149437, 18.07306090799613, 18.55973951718974

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