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Properties

Label 1-291-291.164-r0-0-0
Degree $1$
Conductor $291$
Sign $0.894 + 0.446i$
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 + (0.831 + 0.555i)5-s + (−0.831 + 0.555i)7-s + (0.923 + 0.382i)8-s + (0.195 − 0.980i)10-s + (−0.923 − 0.382i)11-s + (0.555 − 0.831i)13-s + (0.831 + 0.555i)14-s − i·16-s + (−0.555 + 0.831i)17-s + (0.831 + 0.555i)19-s + (−0.980 + 0.195i)20-s + i·22-s + (0.195 + 0.980i)23-s + ⋯
L(s)  = 1  + (−0.382 − 0.923i)2-s + (−0.707 + 0.707i)4-s + (0.831 + 0.555i)5-s + (−0.831 + 0.555i)7-s + (0.923 + 0.382i)8-s + (0.195 − 0.980i)10-s + (−0.923 − 0.382i)11-s + (0.555 − 0.831i)13-s + (0.831 + 0.555i)14-s − i·16-s + (−0.555 + 0.831i)17-s + (0.831 + 0.555i)19-s + (−0.980 + 0.195i)20-s + i·22-s + (0.195 + 0.980i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 291 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.446i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 291 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.446i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(291\)    =    \(3 \cdot 97\)
Sign: $0.894 + 0.446i$
Analytic conductor: \(1.35139\)
Root analytic conductor: \(1.35139\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{291} (164, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 291,\ (0:\ ),\ 0.894 + 0.446i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8442882186 + 0.1989463313i\)
\(L(\frac12)\) \(\approx\) \(0.8442882186 + 0.1989463313i\)
\(L(1)\) \(\approx\) \(0.8278500215 - 0.07921445496i\)
\(L(1)\) \(\approx\) \(0.8278500215 - 0.07921445496i\)

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 \( 1 + (0.831 + 0.555i)T \)
7 \( 1 + (-0.831 + 0.555i)T \)
11 \( 1 + (-0.923 - 0.382i)T \)
13 \( 1 + (0.555 - 0.831i)T \)
17 \( 1 + (-0.555 + 0.831i)T \)
19 \( 1 + (0.831 + 0.555i)T \)
23 \( 1 + (0.195 + 0.980i)T \)
29 \( 1 + (0.195 + 0.980i)T \)
31 \( 1 + (-0.382 + 0.923i)T \)
37 \( 1 + (0.980 + 0.195i)T \)
41 \( 1 + (0.980 - 0.195i)T \)
43 \( 1 + (-0.707 + 0.707i)T \)
47 \( 1 + (-0.707 + 0.707i)T \)
53 \( 1 + (-0.923 + 0.382i)T \)
59 \( 1 + (-0.195 + 0.980i)T \)
61 \( 1 + T \)
67 \( 1 + (0.555 - 0.831i)T \)
71 \( 1 + (0.980 + 0.195i)T \)
73 \( 1 + (-0.707 - 0.707i)T \)
79 \( 1 + (0.382 - 0.923i)T \)
83 \( 1 + (-0.831 - 0.555i)T \)
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

−25.45210724242810874052940916907, −24.62882575538579484378793800626, −23.77209069449670272052494894956, −22.91278384682999431660602595360, −22.04323561405464464501162715617, −20.7604772736538213422944115164, −19.99031144201490312742614671436, −18.718073861455204005889625182088, −18.04525008137540319587622833185, −17.05161823054759123166113692053, −16.26328216491970628459426272290, −15.66864654845201888209635129224, −14.255217122869914900639752803071, −13.43236141089807191743690257038, −12.91546103615428988655819066496, −11.13394537231206252032838113431, −9.844848527337641922540585993030, −9.46291762374454805743664123452, −8.30475135750895978846447818791, −7.08584192232075098987000992140, −6.3101204449805285358478066074, −5.19937447659395789786429452175, −4.241523775092600952926934061289, −2.35934655374673812778173974520, −0.68890207244715152658805197495, 1.49681104113333748948139644817, 2.84275033763672414543534464444, 3.38573158488105812106330082270, 5.24291551067820839211898464130, 6.15541035722766814501877588823, 7.62499997409478075958710233890, 8.7518687406055180729076291100, 9.70805148648297186487078539633, 10.481650033786200527236320087765, 11.27171053303946558773771809025, 12.755034090668974085911258096014, 13.11769213964792097000397090631, 14.198414362734580104007661085937, 15.549470894534726345801056509395, 16.5162347904114992305482124184, 17.875364590082884714196425603761, 18.17945817521961299730321797832, 19.16782645469780553171240816629, 20.05784810020904857999386557913, 21.1660558185818541285815155776, 21.779791294721680782538241988717, 22.540512957262174087054582524, 23.43754644368496521661857542674, 25.03549210288183415112996489461, 25.74989853210612480793791431152

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