Properties

Degree 1
Conductor $ 37 \cdot 109 $
Sign $-0.998 - 0.0550i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  − 2-s + (0.0581 − 0.998i)3-s + 4-s + (−0.918 − 0.396i)5-s + (−0.0581 + 0.998i)6-s + (−0.993 + 0.116i)7-s − 8-s + (−0.993 − 0.116i)9-s + (0.918 + 0.396i)10-s + (−0.918 + 0.396i)11-s + (0.0581 − 0.998i)12-s + (−0.396 + 0.918i)13-s + (0.993 − 0.116i)14-s + (−0.448 + 0.893i)15-s + 16-s + (−0.5 − 0.866i)17-s + ⋯
L(s,χ)  = 1  − 2-s + (0.0581 − 0.998i)3-s + 4-s + (−0.918 − 0.396i)5-s + (−0.0581 + 0.998i)6-s + (−0.993 + 0.116i)7-s − 8-s + (−0.993 − 0.116i)9-s + (0.918 + 0.396i)10-s + (−0.918 + 0.396i)11-s + (0.0581 − 0.998i)12-s + (−0.396 + 0.918i)13-s + (0.993 − 0.116i)14-s + (−0.448 + 0.893i)15-s + 16-s + (−0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.998 - 0.0550i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.998 - 0.0550i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4033\)    =    \(37 \cdot 109\)
\( \varepsilon \)  =  $-0.998 - 0.0550i$
motivic weight  =  \(0\)
character  :  $\chi_{4033} (2719, \cdot )$
Sato-Tate  :  $\mu(108)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4033,\ (0:\ ),\ -0.998 - 0.0550i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.007547074161 - 0.2740013688i$
$L(\frac12,\chi)$  $\approx$  $0.007547074161 - 0.2740013688i$
$L(\chi,1)$  $\approx$  0.4021434948 - 0.1736608184i
$L(1,\chi)$  $\approx$  0.4021434948 - 0.1736608184i

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

−18.91643006735929474078271138457, −18.22378176233882693448009401298, −17.21492731197600440083216476002, −16.78945597681270554651743657663, −16.0190900897442952917029043041, −15.46564810987805046390757214720, −15.22916101815954990417368877151, −14.39544466867586541038727494836, −13.179554561525526684075931410942, −12.50930767290347153664620207677, −11.673090566890523693419385452607, −10.83841888774674678457246195044, −10.54199958919962746473181558038, −9.91901885796576392493193702161, −9.15593312216452028641644488693, −8.39642900348239480637528838201, −7.769237484835341786880354859157, −7.20981307859026448720847017222, −5.905913597451051288622843839, −5.8055054707245769713026080949, −4.34762319767963093839107064387, −3.61853888770771248697547076880, −2.95331325304844515535680090037, −2.40382268824325776376608124929, −0.71510726903063743014352000957, 0.19156952688718932562194617885, 0.90009594218055667672666396244, 2.167474473498605638716782098497, 2.665606768301914389769989348398, 3.4581118703470652723685292051, 4.73126325276159259536561926441, 5.53809431649954582835644519355, 6.73828751250738205558610853419, 7.0108146410544953201858854676, 7.48909423558296350463214880723, 8.480362785277345353955225636572, 9.0271002945050256184242940970, 9.47834499789648402218579318029, 10.7721255563260988194850494021, 11.12190188266321613211484903538, 12.07008477918570426383357916200, 12.54114621950228289963359646397, 13.006206656697407773254129319507, 14.03925071584428437713829009437, 14.92422052630551254886973450675, 15.73090037886593431244246839808, 16.13129138066990495208183549966, 16.83542484313156587136749465837, 17.545549976972593857051761713062, 18.33217503129264299271938762212

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