Properties

Degree 1
Conductor $ 2^{3} \cdot 503 $
Sign $0.949 + 0.314i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.977 + 0.211i)3-s + (−0.0187 − 0.999i)5-s + (0.0437 + 0.999i)7-s + (0.910 + 0.412i)9-s + (0.704 − 0.709i)11-s + (0.686 + 0.726i)13-s + (0.192 − 0.981i)15-s + (−0.915 − 0.401i)17-s + (0.965 + 0.259i)19-s + (−0.168 + 0.985i)21-s + (−0.0812 + 0.996i)23-s + (−0.999 + 0.0375i)25-s + (0.803 + 0.595i)27-s + (0.883 + 0.468i)29-s + (−0.549 − 0.835i)31-s + ⋯
L(s,χ)  = 1  + (0.977 + 0.211i)3-s + (−0.0187 − 0.999i)5-s + (0.0437 + 0.999i)7-s + (0.910 + 0.412i)9-s + (0.704 − 0.709i)11-s + (0.686 + 0.726i)13-s + (0.192 − 0.981i)15-s + (−0.915 − 0.401i)17-s + (0.965 + 0.259i)19-s + (−0.168 + 0.985i)21-s + (−0.0812 + 0.996i)23-s + (−0.999 + 0.0375i)25-s + (0.803 + 0.595i)27-s + (0.883 + 0.468i)29-s + (−0.549 − 0.835i)31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.949 + 0.314i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.949 + 0.314i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4024\)    =    \(2^{3} \cdot 503\)
\( \varepsilon \)  =  $0.949 + 0.314i$
motivic weight  =  \(0\)
character  :  $\chi_{4024} (13, \cdot )$
Sato-Tate  :  $\mu(502)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4024,\ (0:\ ),\ 0.949 + 0.314i)$
$L(\chi,\frac{1}{2})$  $\approx$  $3.001580693 + 0.4838157464i$
$L(\frac12,\chi)$  $\approx$  $3.001580693 + 0.4838157464i$
$L(\chi,1)$  $\approx$  1.673129416 + 0.09031770275i
$L(1,\chi)$  $\approx$  1.673129416 + 0.09031770275i

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

−18.21523932827118753139168172344, −17.984351981475944058073635577185, −17.348363202633471303973703699557, −16.15533529409782970036775432727, −15.65793106656399381777414801353, −14.815488316324415215903099670313, −14.369582074436062814871169206200, −13.807970733231468274701158362196, −13.12600707927682179008086382312, −12.4544909834604405342808653679, −11.46999980309975573224096158217, −10.628362902703552686286321385514, −10.28885752923548888757181007041, −9.37402515467833946762256815969, −8.75156437147618338612284877495, −7.65925676038203414065644425734, −7.46721811614168082735447195222, −6.57862605327713514446518680983, −6.10087601612174940635357641097, −4.53006067288512064314641299396, −4.11788377961882228682041529635, −3.25426599790771912917064989237, −2.669844735180692209538614216452, −1.71609349567626159216879106365, −0.861211225724474733547352534571, 1.049496285434852859773848097947, 1.74945676523870756079114615336, 2.58293910669247507522998826951, 3.52528091061468832154230044059, 4.128096656743890452608413629218, 4.96636684931921426027078170997, 5.71681143477377268433006267807, 6.52432584544672521739278699977, 7.52808567484626944313430704330, 8.304608499011675529670935596847, 8.91586832589858222867711454431, 9.21317168312758606515955905951, 9.84446004465352392695459085282, 11.127225749693702761006816835965, 11.67806589940595145126210137387, 12.34598507887378276280145706752, 13.25140160165281454532474179213, 13.73412369288999499028577301657, 14.26908939263367660882712233441, 15.33050780484925293331405394865, 15.68290520018375705280093982878, 16.35512723415575980684899115203, 16.921582225092776635013017941998, 18.11896409375546395260175616505, 18.45341447892479466769197071071

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