Properties

Label 1-368-368.19-r0-0-0
Degree $1$
Conductor $368$
Sign $-0.303 - 0.952i$
Analytic cond. $1.70898$
Root an. cond. $1.70898$
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  + (−0.540 − 0.841i)3-s + (−0.909 + 0.415i)5-s + (0.959 − 0.281i)7-s + (−0.415 + 0.909i)9-s + (−0.755 + 0.654i)11-s + (0.281 − 0.959i)13-s + (0.841 + 0.540i)15-s + (0.142 − 0.989i)17-s + (0.989 − 0.142i)19-s + (−0.755 − 0.654i)21-s + (0.654 − 0.755i)25-s + (0.989 − 0.142i)27-s + (−0.989 − 0.142i)29-s + (−0.841 − 0.540i)31-s + (0.959 + 0.281i)33-s + ⋯
L(s)  = 1  + (−0.540 − 0.841i)3-s + (−0.909 + 0.415i)5-s + (0.959 − 0.281i)7-s + (−0.415 + 0.909i)9-s + (−0.755 + 0.654i)11-s + (0.281 − 0.959i)13-s + (0.841 + 0.540i)15-s + (0.142 − 0.989i)17-s + (0.989 − 0.142i)19-s + (−0.755 − 0.654i)21-s + (0.654 − 0.755i)25-s + (0.989 − 0.142i)27-s + (−0.989 − 0.142i)29-s + (−0.841 − 0.540i)31-s + (0.959 + 0.281i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.303 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.303 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(368\)    =    \(2^{4} \cdot 23\)
Sign: $-0.303 - 0.952i$
Analytic conductor: \(1.70898\)
Root analytic conductor: \(1.70898\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{368} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 368,\ (0:\ ),\ -0.303 - 0.952i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4528938393 - 0.6193455248i\)
\(L(\frac12)\) \(\approx\) \(0.4528938393 - 0.6193455248i\)
\(L(1)\) \(\approx\) \(0.7252402642 - 0.2779545788i\)
\(L(1)\) \(\approx\) \(0.7252402642 - 0.2779545788i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
23 \( 1 \)
good3 \( 1 + (-0.540 - 0.841i)T \)
5 \( 1 + (-0.909 + 0.415i)T \)
7 \( 1 + (0.959 - 0.281i)T \)
11 \( 1 + (-0.755 + 0.654i)T \)
13 \( 1 + (0.281 - 0.959i)T \)
17 \( 1 + (0.142 - 0.989i)T \)
19 \( 1 + (0.989 - 0.142i)T \)
29 \( 1 + (-0.989 - 0.142i)T \)
31 \( 1 + (-0.841 - 0.540i)T \)
37 \( 1 + (0.909 + 0.415i)T \)
41 \( 1 + (-0.415 - 0.909i)T \)
43 \( 1 + (-0.540 - 0.841i)T \)
47 \( 1 - T \)
53 \( 1 + (-0.281 - 0.959i)T \)
59 \( 1 + (0.281 - 0.959i)T \)
61 \( 1 + (0.540 - 0.841i)T \)
67 \( 1 + (-0.755 - 0.654i)T \)
71 \( 1 + (-0.654 + 0.755i)T \)
73 \( 1 + (0.142 + 0.989i)T \)
79 \( 1 + (-0.959 - 0.281i)T \)
83 \( 1 + (0.909 + 0.415i)T \)
89 \( 1 + (0.841 - 0.540i)T \)
97 \( 1 + (-0.415 - 0.909i)T \)
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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

−24.6541404264210594776786882230, −23.73465830066416931264312427547, −23.482548773871850371737021765906, −22.15463226098050588390416486959, −21.36743869580728391227262924304, −20.762329105908768185121476677156, −19.80439076877965880745784092141, −18.62904600309767033607291677279, −17.87082874280803231983163661641, −16.50884307010159620956758330442, −16.311862931573646465851889531289, −15.13836852995105707154778079737, −14.5446007269893983308957443319, −13.13858382108047451216412417118, −11.92616405123655399878192352651, −11.35525030359712309236945858867, −10.63401053442840757066337667141, −9.27341729082698146732653821240, −8.450138059847676422932438076554, −7.53017093719197457199099289500, −5.981055549780468779811320355635, −5.090846114493672901235811528, −4.23293710502899295275758055609, −3.25327693859627563988002583404, −1.36858983655907789119825419216, 0.544907065140954483008766595847, 2.030482798584004209979774074081, 3.300166485815296670103412205980, 4.80051106315681074776595901381, 5.532110020505700292454559986574, 7.09498478351942452577891972454, 7.58231229067272237391253659805, 8.29976381742286165000190173244, 10.02459805148743193823934876693, 11.13244319013510036825289984219, 11.548053561915362656835984029598, 12.61019095505425778882154006841, 13.505768705884265860391426696717, 14.56631597568479692947265592111, 15.48221487260644376426041478970, 16.429770255271975478674667830346, 17.60763917313993909323140224166, 18.238660892986972527665002536304, 18.80562192422960630979540195020, 20.2445536096549609143841666254, 20.450594153890055712174121084572, 22.13863376736168449689810679094, 22.82333596797470554310560323611, 23.54077712168374587210896988621, 24.19370007916458385318372665296

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