Properties

Degree 1
Conductor 1031
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + 2-s + 3-s + 4-s + 5-s + 6-s − 7-s + 8-s + 9-s + 10-s + 11-s + 12-s + 13-s − 14-s + 15-s + 16-s − 17-s + 18-s − 19-s + 20-s − 21-s + 22-s + 23-s + 24-s + 25-s + 26-s + 27-s − 28-s + ⋯
L(s,χ)  = 1  + 2-s + 3-s + 4-s + 5-s + 6-s − 7-s + 8-s + 9-s + 10-s + 11-s + 12-s + 13-s − 14-s + 15-s + 16-s − 17-s + 18-s − 19-s + 20-s − 21-s + 22-s + 23-s + 24-s + 25-s + 26-s + 27-s − 28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 1031 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & \, \Lambda(\chi,1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 1031 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & \, \Lambda(1-s,\chi) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(1031\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{1031} (1030, \cdot )$
Sato-Tate  :  $\mu(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(1,\ 1031,\ (1:\ ),\ 1)$
$L(\chi,\frac{1}{2})$  $\approx$  $8.632074868$
$L(\frac12,\chi)$  $\approx$  $8.632074868$
$L(\chi,1)$  $\approx$  3.424432297
$L(1,\chi)$  $\approx$  3.424432297

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−21.42543638555137780672971796365, −20.73493804227185617921888243085, −19.947621049242493133390164907167, −19.34899651754004841373857537725, −18.504824767379194072437036309759, −17.272204229819536653149478239499, −16.50804420753397651899959435416, −15.59971809088615957631611434155, −14.98996774495742839982736795264, −14.04075408389827355409628331917, −13.55547200076519402359172975365, −12.95519501725601762435512363994, −12.24938065061276106324961791704, −10.8537807198174235003680097674, −10.28485001826927581931688010107, −9.0666431624907857196483775161, −8.7706391033171722109839017287, −7.13633970817425134081829654227, −6.579648395483394313933279110388, −5.93049129292918572395389638056, −4.61292627816999487508568398384, −3.78059229403184770827031967021, −2.99384159892549382518753739817, −2.11705029197208725078443336089, −1.23567779762651016348635665943, 1.23567779762651016348635665943, 2.11705029197208725078443336089, 2.99384159892549382518753739817, 3.78059229403184770827031967021, 4.61292627816999487508568398384, 5.93049129292918572395389638056, 6.579648395483394313933279110388, 7.13633970817425134081829654227, 8.7706391033171722109839017287, 9.0666431624907857196483775161, 10.28485001826927581931688010107, 10.8537807198174235003680097674, 12.24938065061276106324961791704, 12.95519501725601762435512363994, 13.55547200076519402359172975365, 14.04075408389827355409628331917, 14.98996774495742839982736795264, 15.59971809088615957631611434155, 16.50804420753397651899959435416, 17.272204229819536653149478239499, 18.504824767379194072437036309759, 19.34899651754004841373857537725, 19.947621049242493133390164907167, 20.73493804227185617921888243085, 21.42543638555137780672971796365

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