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Properties

Label 1-2e4-16.3-r1-0-0
Degree $1$
Conductor $16$
Sign $0.382 - 0.923i$
Analytic cond. $1.71943$
Root an. cond. $1.71943$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.382 - 0.923i$
Analytic conductor: \(1.71943\)
Root analytic conductor: \(1.71943\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 16,\ (1:\ ),\ 0.382 - 0.923i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.034220632 - 0.6910441336i\)
\(L(\frac12)\) \(\approx\) \(1.034220632 - 0.6910441336i\)
\(L(1)\) \(\approx\) \(1.026172152 - 0.4250544230i\)
\(L(1)\) \(\approx\) \(1.026172152 - 0.4250544230i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 + T \)
5 \( 1 \)
7 \( 1 - iT \)
11 \( 1 \)
13 \( 1 - iT \)
17 \( 1 \)
19 \( 1 + T \)
23 \( 1 \)
29 \( 1 - T \)
31 \( 1 \)
37 \( 1 + iT \)
41 \( 1 \)
43 \( 1 + iT \)
47 \( 1 \)
53 \( 1 - T \)
59 \( 1 \)
61 \( 1 + T \)
67 \( 1 \)
71 \( 1 - iT \)
73 \( 1 \)
79 \( 1 - iT \)
83 \( 1 \)
89 \( 1 + T \)
97 \( 1 \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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