Properties

Degree 1
Conductor $ 2^{4} $
Sign $0.382 + 0.923i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.382 + 0.923i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 16 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.382 + 0.923i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(16\)    =    \(2^{4}\)
\( \varepsilon \)  =  $0.382 + 0.923i$
motivic weight  =  \(0\)
character  :  $\chi_{16} (11, \cdot )$
Sato-Tate  :  $\mu(4)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 16,\ (1:\ ),\ 0.382 + 0.923i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.034220632 + 0.6910441336i$
$L(\frac12,\chi)$  $\approx$  $1.034220632 + 0.6910441336i$
$L(\chi,1)$  $\approx$  1.026172152 + 0.4250544230i
$L(1,\chi)$  $\approx$  1.026172152 + 0.4250544230i

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

−41.10435922893103128288610544308, −40.56343054924296481332164095922, −38.98239418816734078042574130997, −36.8773893042334511181871341129, −36.243853083102266633789486384, −34.81700297589280402216759773129, −33.32412131715627350233816355356, −31.4961575761160590501466384261, −30.49364920574690538785063011518, −28.8866673667461449447663431348, −27.75377435601106270283031063944, −25.60030431822583729053980869233, −24.32295393554366501969712582159, −23.40495602885311021627056367523, −21.10602533809967547634175422685, −19.76937889891387206085621487717, −18.06219286010072963536486991815, −16.82445624420277850920199241233, −14.52615200128381787765942423484, −12.89312927313998434914606507415, −11.60573201982624604500676880012, −8.91089908173240970351299825011, −7.32389006084205077392864564526, −5.01745981013337413248306330226, −1.58558376470855129400428237197, 3.34621940663383010487516724722, 5.56410941089718972889114454396, 8.11780157119913457272179571068, 10.24986090941885860437089843593, 11.38476129885738015330607200569, 14.17899543196987880915104051642, 15.238424483154038119797584353439, 17.02866369225634600170959314957, 18.70516602154717034912678399067, 20.68218228520702597171400481982, 21.85061764253219008272661542694, 23.19694306478602257079964841932, 25.249610739579538384551623638478, 26.84664899986183171968240634940, 27.535359399349516385974382954457, 29.53422701439338231841300427581, 30.97043604311530505334513565258, 32.46345155287375067876625734070, 33.86001657021730933586480726849, 34.66672417933103563986905652739, 37.1242483515259595832879410422, 37.78294728438758778036278040, 39.24061520220888061125445290225, 40.44992337224810660229123237792, 42.26721837804603158276839961467

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