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Properties

Label 1-291-291.116-r0-0-0
Degree $1$
Conductor $291$
Sign $-0.994 + 0.101i$
Analytic cond. $1.35139$
Root an. cond. $1.35139$
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  + (0.382 + 0.923i)2-s + (−0.707 + 0.707i)4-s + (−0.555 + 0.831i)5-s + (0.555 + 0.831i)7-s + (−0.923 − 0.382i)8-s + (−0.980 − 0.195i)10-s + (0.923 + 0.382i)11-s + (0.831 + 0.555i)13-s + (−0.555 + 0.831i)14-s − i·16-s + (−0.831 − 0.555i)17-s + (−0.555 + 0.831i)19-s + (−0.195 − 0.980i)20-s + i·22-s + (−0.980 + 0.195i)23-s + ⋯
L(s)  = 1  + (0.382 + 0.923i)2-s + (−0.707 + 0.707i)4-s + (−0.555 + 0.831i)5-s + (0.555 + 0.831i)7-s + (−0.923 − 0.382i)8-s + (−0.980 − 0.195i)10-s + (0.923 + 0.382i)11-s + (0.831 + 0.555i)13-s + (−0.555 + 0.831i)14-s − i·16-s + (−0.831 − 0.555i)17-s + (−0.555 + 0.831i)19-s + (−0.195 − 0.980i)20-s + i·22-s + (−0.980 + 0.195i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 291 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.101i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 291 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.101i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(291\)    =    \(3 \cdot 97\)
Sign: $-0.994 + 0.101i$
Analytic conductor: \(1.35139\)
Root analytic conductor: \(1.35139\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{291} (116, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 291,\ (0:\ ),\ -0.994 + 0.101i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.06058987020 + 1.189268905i\)
\(L(\frac12)\) \(\approx\) \(0.06058987020 + 1.189268905i\)
\(L(1)\) \(\approx\) \(0.7044633780 + 0.8436033963i\)
\(L(1)\) \(\approx\) \(0.7044633780 + 0.8436033963i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
97 \( 1 \)
good2 \( 1 + (0.382 + 0.923i)T \)
5 \( 1 + (-0.555 + 0.831i)T \)
7 \( 1 + (0.555 + 0.831i)T \)
11 \( 1 + (0.923 + 0.382i)T \)
13 \( 1 + (0.831 + 0.555i)T \)
17 \( 1 + (-0.831 - 0.555i)T \)
19 \( 1 + (-0.555 + 0.831i)T \)
23 \( 1 + (-0.980 + 0.195i)T \)
29 \( 1 + (-0.980 + 0.195i)T \)
31 \( 1 + (0.382 - 0.923i)T \)
37 \( 1 + (0.195 - 0.980i)T \)
41 \( 1 + (0.195 + 0.980i)T \)
43 \( 1 + (-0.707 + 0.707i)T \)
47 \( 1 + (-0.707 + 0.707i)T \)
53 \( 1 + (0.923 - 0.382i)T \)
59 \( 1 + (0.980 + 0.195i)T \)
61 \( 1 + T \)
67 \( 1 + (0.831 + 0.555i)T \)
71 \( 1 + (0.195 - 0.980i)T \)
73 \( 1 + (-0.707 - 0.707i)T \)
79 \( 1 + (-0.382 + 0.923i)T \)
83 \( 1 + (0.555 - 0.831i)T \)
89 \( 1 + (0.923 + 0.382i)T \)
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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

−24.6268140513540304917732266880, −23.960130209053931557088588048073, −23.28540678643959219994040887869, −22.25363669226271289168696603848, −21.332852438851502758631690411945, −20.27012955177473202534661534844, −20.002331345975052249071334721385, −19.01260202243464941980259472668, −17.74266194418261731466311473006, −17.01156934768828127486868401280, −15.730177721564445496680895799040, −14.732442523829612658123759856155, −13.60560408819807073306697179803, −13.028754984347466190176867217, −11.827674601471365819916646550719, −11.17028859411393390395349958730, −10.23558137143241080349378064087, −8.840584530148960468841025271926, −8.30122899834647147837622828940, −6.63808753147614450365508598442, −5.30660095081553321141258911837, −4.195151936288585559132380964065, −3.674337551056700158611860024708, −1.8374985722566853703968263682, −0.73095523676190624059692063292, 2.14661864966029874757718858824, 3.68079003411126130707938882124, 4.4405258849113345758363251474, 5.92342179744402737819528066678, 6.6156804118996905031725552705, 7.73573057649481907315801592971, 8.6139153299034696337731388683, 9.63178573977792073407734745214, 11.33800176929515597431731735567, 11.84786933015693698438732223357, 13.10901036690627184564791056720, 14.352999269372978245302497070825, 14.76837466590518617111211946989, 15.70725784352714584637743499187, 16.53612756840966858093934064306, 17.85048680883867206495406579389, 18.33977112525666800720608342888, 19.34224777810381454494962387414, 20.72489342694322720885510383438, 21.762907183101898354554046956048, 22.45772742225759995714536148982, 23.19847316887246288276457339050, 24.18221348990627219902065821986, 24.950066983887870906131596141326, 25.83210528019108047044073618171

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