Properties

Degree 1
Conductor $ 2^{2} \cdot 751 $
Sign $-0.984 - 0.175i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.309 + 0.951i)3-s + (0.309 + 0.951i)5-s + (0.309 + 0.951i)7-s + (−0.809 + 0.587i)9-s + 11-s + (−0.809 − 0.587i)13-s + (−0.809 + 0.587i)15-s + (−0.309 + 0.951i)17-s + (0.809 + 0.587i)19-s + (−0.809 + 0.587i)21-s + (0.809 − 0.587i)23-s + (−0.809 + 0.587i)25-s + (−0.809 − 0.587i)27-s + (0.809 − 0.587i)29-s + (−0.809 + 0.587i)31-s + ⋯
L(s,χ)  = 1  + (0.309 + 0.951i)3-s + (0.309 + 0.951i)5-s + (0.309 + 0.951i)7-s + (−0.809 + 0.587i)9-s + 11-s + (−0.809 − 0.587i)13-s + (−0.809 + 0.587i)15-s + (−0.309 + 0.951i)17-s + (0.809 + 0.587i)19-s + (−0.809 + 0.587i)21-s + (0.809 − 0.587i)23-s + (−0.809 + 0.587i)25-s + (−0.809 − 0.587i)27-s + (0.809 − 0.587i)29-s + (−0.809 + 0.587i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 3004 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.984 - 0.175i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 3004 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.984 - 0.175i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(3004\)    =    \(2^{2} \cdot 751\)
\( \varepsilon \)  =  $-0.984 - 0.175i$
motivic weight  =  \(0\)
character  :  $\chi_{3004} (671, \cdot )$
Sato-Tate  :  $\mu(10)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 3004,\ (0:\ ),\ -0.984 - 0.175i)$
$L(\chi,\frac{1}{2})$  $\approx$  $-0.1514154275 + 1.708170481i$
$L(\frac12,\chi)$  $\approx$  $-0.1514154275 + 1.708170481i$
$L(\chi,1)$  $\approx$  0.9002786215 + 0.8136864558i
$L(1,\chi)$  $\approx$  0.9002786215 + 0.8136864558i

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

−18.77860348021087986351827019408, −17.70255873659377662232161343149, −17.483353365728492612778333077723, −16.73898164819053482673588791685, −16.15861733914610059915766948862, −14.96848979280091170882458758307, −14.26100511566581888795110397560, −13.62044462276245091071255128349, −13.33297155698584811660484007469, −12.23356982030705561538892615653, −11.85073335776930134661600052616, −11.120604251514263693168586693868, −9.937676783510183344713768650, −9.10590379932719583097405481348, −8.87930921425584413005746547588, −7.708623759158792224252214542955, −7.12148066381867545558880293871, −6.67152640784505011230447591852, −5.45207199574168103839049382659, −4.84066545948238189217227065254, −3.95212459823676061241048160024, −3.004888526208066410589569603377, −1.913795512723744529503185836687, −1.29424910099381325050650926904, −0.48434814998100535195238292085, 1.57874762009434103543947645411, 2.44335756611108010563966420843, 3.15324635203030411337668326409, 3.81073298368474393891853960460, 4.89285903704704337426975837168, 5.49024202612512581325982196209, 6.3172045718785140446331489548, 7.09129683062643929850216502801, 8.20772295643662277099409602377, 8.68454467821229724073988574053, 9.61460880734631996521753755784, 10.09360423859577822381176809515, 10.80535561750563064140480263216, 11.6100248752716505371716635168, 12.1544779860000565935686136254, 13.22651323938167014801154230030, 14.17329508285072092893945235069, 14.71139869032022257431624379863, 15.03005329504531327648709111127, 15.73089712303682710244503504623, 16.67895926359281061343504267827, 17.34856264722769884265973522829, 17.94275640875079174934537412362, 18.852786816587639888148818449110, 19.42242981680581203974971348516

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