Properties

Degree 1
Conductor $ 2^{2} \cdot 251 $
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes
Analytic rank 0

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  − 3-s + 5-s − 7-s + 9-s + 11-s + 13-s − 15-s + 17-s + 19-s + 21-s − 23-s + 25-s − 27-s − 29-s − 31-s − 33-s − 35-s − 37-s − 39-s + 41-s + 43-s + 45-s + 47-s + 49-s − 51-s − 53-s + 55-s + ⋯
L(s,χ)  = 1  − 3-s + 5-s − 7-s + 9-s + 11-s + 13-s − 15-s + 17-s + 19-s + 21-s − 23-s + 25-s − 27-s − 29-s − 31-s − 33-s − 35-s − 37-s − 39-s + 41-s + 43-s + 45-s + 47-s + 49-s − 51-s − 53-s + 55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 1004 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & \, \Lambda(\chi,1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 1004 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & \, \Lambda(1-s,\chi) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(1004\)    =    \(2^{2} \cdot 251\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{1004} (1003, \cdot )$
Sato-Tate  :  $\mu(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(1,\ 1004,\ (0:\ ),\ 1)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.350956357$
$L(\frac12,\chi)$  $\approx$  $1.350956357$
$L(\chi,1)$  $\approx$  0.9979297720
$L(1,\chi)$  $\approx$  0.9979297720

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

−21.87610023389867489719942655516, −20.99049019612420743785098681355, −20.20821249196175874171907908370, −19.06202924110475618624612638626, −18.423876092065867803989808159162, −17.7100165383636279995115619778, −16.862840218712791653142513621832, −16.342777327615641720704765172, −15.65341221762891875233794411992, −14.32885796717620037679812023550, −13.69409826962796146746989180930, −12.75973647768256395122504262490, −12.188883898582944180466029196811, −11.19711624793667349582443692694, −10.37458555676736235232750601799, −9.580657603527533003195615303938, −9.09018656084749824320700545676, −7.521102083885829897671875712270, −6.67612869494582916600054283854, −5.82804481703581364904424565926, −5.59086586079126809946204098316, −4.09307149798994591774613990473, −3.31913511116474216131256857300, −1.814251052475068643603607623679, −0.94178475154299151351805066607, 0.94178475154299151351805066607, 1.814251052475068643603607623679, 3.31913511116474216131256857300, 4.09307149798994591774613990473, 5.59086586079126809946204098316, 5.82804481703581364904424565926, 6.67612869494582916600054283854, 7.521102083885829897671875712270, 9.09018656084749824320700545676, 9.580657603527533003195615303938, 10.37458555676736235232750601799, 11.19711624793667349582443692694, 12.188883898582944180466029196811, 12.75973647768256395122504262490, 13.69409826962796146746989180930, 14.32885796717620037679812023550, 15.65341221762891875233794411992, 16.342777327615641720704765172, 16.862840218712791653142513621832, 17.7100165383636279995115619778, 18.423876092065867803989808159162, 19.06202924110475618624612638626, 20.20821249196175874171907908370, 20.99049019612420743785098681355, 21.87610023389867489719942655516

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