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Properties

Label 1-368-368.197-r0-0-0
Degree $1$
Conductor $368$
Sign $0.189 - 0.981i$
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  + (0.909 − 0.415i)3-s + (0.755 − 0.654i)5-s + (−0.841 − 0.540i)7-s + (0.654 − 0.755i)9-s + (0.989 − 0.142i)11-s + (−0.540 − 0.841i)13-s + (0.415 − 0.909i)15-s + (−0.959 + 0.281i)17-s + (−0.281 + 0.959i)19-s + (−0.989 − 0.142i)21-s + (0.142 − 0.989i)25-s + (0.281 − 0.959i)27-s + (0.281 + 0.959i)29-s + (0.415 − 0.909i)31-s + (0.841 − 0.540i)33-s + ⋯
L(s)  = 1  + (0.909 − 0.415i)3-s + (0.755 − 0.654i)5-s + (−0.841 − 0.540i)7-s + (0.654 − 0.755i)9-s + (0.989 − 0.142i)11-s + (−0.540 − 0.841i)13-s + (0.415 − 0.909i)15-s + (−0.959 + 0.281i)17-s + (−0.281 + 0.959i)19-s + (−0.989 − 0.142i)21-s + (0.142 − 0.989i)25-s + (0.281 − 0.959i)27-s + (0.281 + 0.959i)29-s + (0.415 − 0.909i)31-s + (0.841 − 0.540i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.189 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.189 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(368\)    =    \(2^{4} \cdot 23\)
Sign: $0.189 - 0.981i$
Analytic conductor: \(1.70898\)
Root analytic conductor: \(1.70898\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{368} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 368,\ (0:\ ),\ 0.189 - 0.981i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.405717007 - 1.160803943i\)
\(L(\frac12)\) \(\approx\) \(1.405717007 - 1.160803943i\)
\(L(1)\) \(\approx\) \(1.358903789 - 0.5441128370i\)
\(L(1)\) \(\approx\) \(1.358903789 - 0.5441128370i\)

Euler product

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

−25.01987578158621287739768742897, −24.3393867777226155573342305629, −22.76508166819382895123304664778, −21.94329340883971719955774361453, −21.651183067577520454314755908126, −20.44649455182050571042295988729, −19.35940207938311011289839718145, −19.08334632681169080898890704463, −17.791734070892531320490562683857, −16.87432960254634438727088619780, −15.69780717185545728132582142324, −15.05885668954880496691398899230, −14.02887718465064550390760231583, −13.52420468561186613343346426872, −12.31632712327542367621149124909, −11.1182126796522248887058042904, −9.947009007793521564328815967764, −9.342868134902790062018618015036, −8.664948530025925067991093595525, −6.95752443749279907771928095226, −6.54662234596597668156321811605, −4.994528721115750324892372145848, −3.831214560469570794345732451324, −2.69616547418454278699219394618, −1.9945664512231237306515279254, 1.03915527015015775638379917454, 2.209353947262063543747922278673, 3.41768292996208215749508198069, 4.43016761191884317566442426638, 5.99236067033527669208750395951, 6.76434140248529008510754101797, 7.93914268443199036094229586548, 8.95708968384180150078151353412, 9.629117190195769747621667211409, 10.5331359865770305584441466405, 12.26306935793497372422297593657, 12.84343982579866069702215932801, 13.66278629285395845429319771422, 14.41737990108617003623432383137, 15.46613197661458791593143554076, 16.61510409382012993064505636996, 17.33951259526507759660004716646, 18.3210233864108832177465582283, 19.541847144297299608339207078046, 19.927312554376608481323925671338, 20.75115374117531289907424441173, 21.8016650015279950671299149458, 22.649682698717597493024989714773, 23.82019061333077879247662177980, 24.807391244678515879723646760178

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