Properties

Degree 2
Conductor 389
Sign $1$
Self-dual yes
Motivic weight 1

Origins

Related objects

Downloads

Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 2·3-s + 2·4-s − 3·5-s + 4·6-s − 5·7-s + 9-s + 6·10-s − 4·11-s − 4·12-s − 3·13-s + 10·14-s + 6·15-s − 4·16-s − 6·17-s − 2·18-s + 5·19-s − 6·20-s + 10·21-s + 8·22-s − 4·23-s + 4·25-s + 6·26-s + 4·27-s − 10·28-s − 6·29-s − 12·30-s + ⋯
L(s)  = 1  − 1.414·2-s − 1.154·3-s + 4-s − 1.341·5-s + 1.632·6-s − 1.889·7-s + 0.333·9-s + 1.897·10-s − 1.206·11-s − 1.154·12-s − 0.832·13-s + 2.672·14-s + 1.549·15-s − 16-s − 1.455·17-s − 0.471·18-s + 1.147·19-s − 1.341·20-s + 2.182·21-s + 1.705·22-s − 0.834·23-s + 0.8·25-s + 1.176·26-s + 0.769·27-s − 1.889·28-s − 1.114·29-s − 2.190·30-s + ⋯

Functional equation

\[\begin{align} \Lambda(s)=\mathstrut & 389 ^{s/2} \Gamma_{\C}(s) \cdot L(s)\cr =\mathstrut & \Lambda(2-s) \end{align} \]
\[\begin{align} \Lambda(s)=\mathstrut & 389 ^{s/2} \Gamma_{\C}(s+0.5) \cdot L(s)\cr =\mathstrut & \Lambda(1-s) \end{align} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(389\)
\( \varepsilon \)  =  $1$
weight  =  1
character  :  $\chi_{389} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(2,\ 389,\ (\ :1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 389$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p = 389$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad389$1-T$
good2$1+2T+2T^{2}$
3$1+2T+3T^{2}$
5$1+3T+5T^{2}$
7$1+5T+7T^{2}$
11$1+4T+11T^{2}$
13$1+3T+13T^{2}$
17$1+6T+17T^{2}$
19$1-5T+19T^{2}$
23$1+4T+23T^{2}$
29$1+6T+29T^{2}$
31$1-4T+31T^{2}$
37$1+8T+37T^{2}$
41$1+3T+41T^{2}$
43$1-12T+43T^{2}$
47$1+2T+47T^{2}$
53$1+6T+53T^{2}$
59$1-3T+59T^{2}$
61$1+8T+61T^{2}$
67$1+5T+67T^{2}$
71$1+10T+71T^{2}$
73$1+7T+73T^{2}$
79$1+13T+79T^{2}$
83$1+12T+83T^{2}$
89$1+8T+89T^{2}$
97$1+9T+97T^{2}$
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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

−19.86122175437047, −19.27344364744404, −18.69090393427191, −17.86770328341567, −17.17988299144824, −16.25006992620260, −15.91158602586977, −15.50561847327667, −13.62485367774964, −12.66721374574122, −11.93352732788421, −11.11093553880680, −10.35143331288150, −9.633078802184913, −8.633205244563326, −7.474907495785431, −6.985966652828689, −5.793402633928365, −4.416896083665258, −2.876099071260465, 0, 0, 2.876099071260465, 4.416896083665258, 5.793402633928365, 6.985966652828689, 7.474907495785431, 8.633205244563326, 9.633078802184913, 10.35143331288150, 11.11093553880680, 11.93352732788421, 12.66721374574122, 13.62485367774964, 15.50561847327667, 15.91158602586977, 16.25006992620260, 17.17988299144824, 17.86770328341567, 18.69090393427191, 19.27344364744404, 19.86122175437047

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