Properties

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

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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

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & \, \Lambda(\chi,1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 4 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & \, \Lambda(1-s,\chi) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4\)    =    \(2^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{4} (3, \cdot )$
Sato-Tate  :  $\mu(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(1,\ 4,\ (1:\ ),\ 1)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.6676914571$
$L(\frac12,\chi)$  $\approx$  $0.6676914571$
$L(\chi,1)$  $\approx$  0.7853981633
$L(1,\chi)$  $\approx$  0.7853981633

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

−58.1167071106739179772623675469, −56.934374055202296886801711961125, −55.267543584699224846718259656915, −52.76882076780472926503507657877, −51.68609345287052843953381111031, −49.723129323782586066569570880970, −47.74156228093914125078134734303, −45.59958439679156674593770229335, −44.6178910586623033934820457250, −41.80708462000456233715752189724, −40.32267406669054418034439436231, −38.511923141718691293776504688065, −36.14288045830313783056581447010, −34.19995750921314691304479547000, −32.59218652711715513081519404895, −29.65638401459315272180990696821, −28.35963434302532778565160794178, −25.728756425088727567265088674277, −23.27837652045953153181955888634, −21.45061134398346049720094838629, −18.29199319612353483852600427759, −16.34260710458722219497686148345, −12.98809801231242250745310978956, −10.2437703041665545521377574791, −6.02094890469759665490251152161, 6.02094890469759665490251152161, 10.2437703041665545521377574791, 12.98809801231242250745310978956, 16.34260710458722219497686148345, 18.29199319612353483852600427759, 21.45061134398346049720094838629, 23.27837652045953153181955888634, 25.728756425088727567265088674277, 28.35963434302532778565160794178, 29.65638401459315272180990696821, 32.59218652711715513081519404895, 34.19995750921314691304479547000, 36.14288045830313783056581447010, 38.511923141718691293776504688065, 40.32267406669054418034439436231, 41.80708462000456233715752189724, 44.6178910586623033934820457250, 45.59958439679156674593770229335, 47.74156228093914125078134734303, 49.723129323782586066569570880970, 51.68609345287052843953381111031, 52.76882076780472926503507657877, 55.267543584699224846718259656915, 56.934374055202296886801711961125, 58.1167071106739179772623675469

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