Properties

Label 1-368-368.301-r0-0-0
Degree $1$
Conductor $368$
Sign $-0.989 - 0.147i$
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  + (−0.281 − 0.959i)3-s + (0.540 − 0.841i)5-s + (0.142 + 0.989i)7-s + (−0.841 + 0.540i)9-s + (−0.909 − 0.415i)11-s + (−0.989 − 0.142i)13-s + (−0.959 − 0.281i)15-s + (−0.654 − 0.755i)17-s + (−0.755 − 0.654i)19-s + (0.909 − 0.415i)21-s + (−0.415 − 0.909i)25-s + (0.755 + 0.654i)27-s + (0.755 − 0.654i)29-s + (−0.959 − 0.281i)31-s + (−0.142 + 0.989i)33-s + ⋯
L(s)  = 1  + (−0.281 − 0.959i)3-s + (0.540 − 0.841i)5-s + (0.142 + 0.989i)7-s + (−0.841 + 0.540i)9-s + (−0.909 − 0.415i)11-s + (−0.989 − 0.142i)13-s + (−0.959 − 0.281i)15-s + (−0.654 − 0.755i)17-s + (−0.755 − 0.654i)19-s + (0.909 − 0.415i)21-s + (−0.415 − 0.909i)25-s + (0.755 + 0.654i)27-s + (0.755 − 0.654i)29-s + (−0.959 − 0.281i)31-s + (−0.142 + 0.989i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.989 - 0.147i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.989 - 0.147i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(368\)    =    \(2^{4} \cdot 23\)
Sign: $-0.989 - 0.147i$
Analytic conductor: \(1.70898\)
Root analytic conductor: \(1.70898\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{368} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 368,\ (0:\ ),\ -0.989 - 0.147i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04619929675 - 0.6249379347i\)
\(L(\frac12)\) \(\approx\) \(0.04619929675 - 0.6249379347i\)
\(L(1)\) \(\approx\) \(0.6764072711 - 0.3962611618i\)
\(L(1)\) \(\approx\) \(0.6764072711 - 0.3962611618i\)

Euler product

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

−25.47357255929590289350756813147, −23.86707175933322738210116980458, −23.29743895439572782469122379042, −22.2727496889043623519467833955, −21.710346783904303798591252769385, −20.79209419951041665248798702893, −20.02124029422507509101470980263, −18.90425597881112556993285410610, −17.65920781794610872856487184168, −17.25118245635559453493596981986, −16.25218386483808689415433973495, −15.10310822103930776887659063962, −14.576783624702324438001499938394, −13.5902273772101524106637683111, −12.4343642079167247081069987395, −11.11655060431899249105888152593, −10.31166966333030492892167017780, −10.07804065211952090798517014294, −8.67425618914270704172649141088, −7.37791880187454832826714604931, −6.46424593715623820341047458371, −5.27510140221458640406938224800, −4.33839138506695720392734204039, −3.25192036634378714749002909563, −2.02222991609427374121306019393, 0.355859287550925900794639667930, 2.05679461978366384688348806910, 2.62201495614639669852797222082, 4.83455072132207825979684263862, 5.43987669921887027998951292802, 6.41573707039338554912671613561, 7.63524323471770686959072792405, 8.567984077435629742136447814292, 9.34732309808793476165691197058, 10.736174019560079237321291283561, 11.80916918444240333375377507948, 12.59608017253778179397194774853, 13.22910738739935816592642201490, 14.160616975943232533610150443888, 15.40820272090447155345900154327, 16.348002302009136837289847048744, 17.42213226426897772903398086670, 17.93151647872707146649672179770, 18.89804265745374641349676209178, 19.71468948114696783709157588496, 20.760794025265950033526757363612, 21.685496396839397841892419835837, 22.42389934830094255727600969281, 23.72307102735106908979373539248, 24.24174262000412775248109385712

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