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Properties

Label 1-291-291.152-r0-0-0
Degree $1$
Conductor $291$
Sign $-0.915 - 0.402i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 291 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.915 - 0.402i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 291 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.915 - 0.402i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(291\)    =    \(3 \cdot 97\)
Sign: $-0.915 - 0.402i$
Analytic conductor: \(1.35139\)
Root analytic conductor: \(1.35139\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{291} (152, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 291,\ (0:\ ),\ -0.915 - 0.402i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.08154453634 + 0.3880975910i\)
\(L(\frac12)\) \(\approx\) \(-0.08154453634 + 0.3880975910i\)
\(L(1)\) \(\approx\) \(0.4665440659 + 0.3772045698i\)
\(L(1)\) \(\approx\) \(0.4665440659 + 0.3772045698i\)

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

−24.92566305670670726940874612362, −23.78248571904125642292647604325, −23.32900057742546328314706153838, −22.05828032568506169900407985727, −20.92274280524702279613883885822, −20.65252779946717556745625585154, −19.49061978009174672181093830885, −18.84792196514222573772796535531, −17.79551145942335649661861796407, −16.81958778787515121177637356461, −16.097311105412898384414003366964, −14.65201463631624277655311530010, −13.6534055278493993782701701214, −12.65085630918684954048119937998, −11.72208982847907603177682401374, −11.03541829473352676697424798065, −10.02847083394683745655057438448, −8.764444213503092138702072040634, −8.053818220700520688265702415977, −7.148646095911818373804458073980, −4.96095151714934141557331741834, −4.44725674220820776835874372879, −3.13305162240509303474774860150, −1.77918260228464876138250515929, −0.29621568095319150201492744744, 1.88412282680996503965193497185, 3.57279971747909405515973300431, 4.92438740769633500639020279616, 5.72357499813593249377364981734, 7.12533416791970586049800046019, 7.93107321048081920873912233997, 8.4959475946297673528988474467, 10.06510496153383955901503763757, 10.73866737207924001325520332434, 12.0394594269670579907710838183, 13.09947862824663156265001383774, 14.54644845851740911047208440617, 15.04347406696396685315151169537, 15.65971824227509756624782422829, 16.89558479198411767756080171022, 17.822274856043089304954506277193, 18.56843363015263738695935202747, 19.31997513878229592665884463120, 20.43093911964197863919332296210, 21.70635962710804117719194508340, 22.62431097003192532728029259779, 23.655419395508212120966168019486, 23.969878949602716473955050723379, 25.278492086562441697057723547029, 25.858449258857906884424273251807

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