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Properties

Label 1-293-293.65-r0-0-0
Degree $1$
Conductor $293$
Sign $0.840 - 0.541i$
Analytic cond. $1.36068$
Root an. cond. $1.36068$
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  + (−0.890 + 0.455i)2-s + (−0.474 − 0.880i)3-s + (0.584 − 0.811i)4-s + (−0.976 + 0.213i)5-s + (0.823 + 0.566i)6-s + (−0.954 + 0.296i)7-s + (−0.150 + 0.988i)8-s + (−0.548 + 0.835i)9-s + (0.772 − 0.635i)10-s + (0.512 − 0.858i)11-s + (−0.991 − 0.128i)12-s + (0.357 + 0.933i)13-s + (0.714 − 0.699i)14-s + (0.651 + 0.758i)15-s + (−0.317 − 0.948i)16-s + (−0.890 + 0.455i)17-s + ⋯
L(s)  = 1  + (−0.890 + 0.455i)2-s + (−0.474 − 0.880i)3-s + (0.584 − 0.811i)4-s + (−0.976 + 0.213i)5-s + (0.823 + 0.566i)6-s + (−0.954 + 0.296i)7-s + (−0.150 + 0.988i)8-s + (−0.548 + 0.835i)9-s + (0.772 − 0.635i)10-s + (0.512 − 0.858i)11-s + (−0.991 − 0.128i)12-s + (0.357 + 0.933i)13-s + (0.714 − 0.699i)14-s + (0.651 + 0.758i)15-s + (−0.317 − 0.948i)16-s + (−0.890 + 0.455i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 293 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.840 - 0.541i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 293 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.840 - 0.541i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(293\)
Sign: $0.840 - 0.541i$
Analytic conductor: \(1.36068\)
Root analytic conductor: \(1.36068\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{293} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 293,\ (0:\ ),\ 0.840 - 0.541i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4161037821 - 0.1224224841i\)
\(L(\frac12)\) \(\approx\) \(0.4161037821 - 0.1224224841i\)
\(L(1)\) \(\approx\) \(0.4789530499 - 0.03421916366i\)
\(L(1)\) \(\approx\) \(0.4789530499 - 0.03421916366i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad293 \( 1 \)
good2 \( 1 + (-0.890 + 0.455i)T \)
3 \( 1 + (-0.474 - 0.880i)T \)
5 \( 1 + (-0.976 + 0.213i)T \)
7 \( 1 + (-0.954 + 0.296i)T \)
11 \( 1 + (0.512 - 0.858i)T \)
13 \( 1 + (0.357 + 0.933i)T \)
17 \( 1 + (-0.890 + 0.455i)T \)
19 \( 1 + (0.651 + 0.758i)T \)
23 \( 1 + (0.823 - 0.566i)T \)
29 \( 1 + (0.192 - 0.981i)T \)
31 \( 1 + (-0.976 - 0.213i)T \)
37 \( 1 + (-0.798 + 0.601i)T \)
41 \( 1 + (-0.618 - 0.785i)T \)
43 \( 1 + (0.436 - 0.899i)T \)
47 \( 1 + (-0.0645 - 0.997i)T \)
53 \( 1 + (0.512 - 0.858i)T \)
59 \( 1 + (0.966 - 0.255i)T \)
61 \( 1 + (0.941 + 0.337i)T \)
67 \( 1 + (0.869 + 0.493i)T \)
71 \( 1 + (-0.999 + 0.0430i)T \)
73 \( 1 + (0.584 + 0.811i)T \)
79 \( 1 + (0.941 + 0.337i)T \)
83 \( 1 + (0.996 + 0.0859i)T \)
89 \( 1 + (0.941 - 0.337i)T \)
97 \( 1 + (0.0215 - 0.999i)T \)
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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

−25.86373433433792113547768158696, −24.93209468703762174943512646363, −23.504987754450713668552308844, −22.60373513324275527293582346306, −22.062525754644186955124892061099, −20.65562850110070227813543342387, −20.0350156772201101219863337601, −19.53332461307084042955862357199, −18.09891788910611623659977455130, −17.379174045538511105201800933226, −16.291079469439964491513429841, −15.810100081954700190577408963247, −15.01523084061649052281617303480, −13.06646792273059236741923910501, −12.24938694486940324411760055279, −11.261243643260388211350990755805, −10.59567492550996152496426319231, −9.43733905566911835300833475179, −8.92063351569551794641506654169, −7.43225632511960325505503267603, −6.644085513905767229787269449442, −4.94478791400300582340670098792, −3.75156118548994784629436594349, −3.03339796244865765997839430518, −0.8287508942255349209955633608, 0.61689989191219786225109842528, 2.15809704165263629126133348144, 3.66633749979129581665222856614, 5.50210037571539346961740905103, 6.597026560800792704383698357360, 6.98113114445829159248750856555, 8.31720221453820234231054114433, 8.93841231392137005612729064024, 10.431633310337769567969423335471, 11.42697588965372073250460696215, 11.991781005455610112744902292878, 13.33205463634607495970066363070, 14.42252444672198248970987919785, 15.62973841938037515403004540342, 16.396902915325331872959420940545, 17.026640021368696638295595087416, 18.35045014781143200466084866268, 19.0733600710316422740806277330, 19.29281307760576626390507491253, 20.42953724582858652780387346360, 22.14442286110158546878835633768, 22.90140453933282887904936193889, 23.84308941857982404969896124257, 24.423020702579850855177445117194, 25.334626520796428986547362518262

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