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Properties

Label 1-291-291.245-r0-0-0
Degree $1$
Conductor $291$
Sign $0.977 - 0.213i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 291 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.213i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 291 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.213i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(291\)    =    \(3 \cdot 97\)
Sign: $0.977 - 0.213i$
Analytic conductor: \(1.35139\)
Root analytic conductor: \(1.35139\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{291} (245, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 291,\ (0:\ ),\ 0.977 - 0.213i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.433141829 - 0.1545574920i\)
\(L(\frac12)\) \(\approx\) \(1.433141829 - 0.1545574920i\)
\(L(1)\) \(\approx\) \(1.218762978 - 0.2700774730i\)
\(L(1)\) \(\approx\) \(1.218762978 - 0.2700774730i\)

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

−25.3614695056156557477906559314, −24.64373556679611065973608824804, −23.8592718432157294650476009627, −22.852675990910609413795105267524, −22.165816469683735288084342405932, −21.14010433832777971901286311556, −20.16332651635887080590500957699, −19.22904421935057890377247677218, −17.6729700847010653838792453983, −17.10786508094331787835054751272, −16.5549052659579875417475167543, −15.40162358525435442000119620065, −14.42209264760450482536455830074, −13.50702855762997852391234294449, −12.7857937952693601717941877939, −11.90984472235887036040473715168, −10.05617364785452785647137598788, −9.41605706625229491424898726156, −8.273382782004097562629628727402, −7.179348963195987084661128617504, −6.30718302977139717278628559232, −5.13222839700954711629306084745, −4.30963728726370970645911649454, −3.02720824845447158551064026805, −0.96506187386214190054966696277, 1.51954529089152718011368222893, 2.776176292676472124844526037551, 3.455487421373206314884868818048, 5.049222143180489619887598528686, 5.99583090665308534878743942082, 6.99146430029407578656811136206, 8.82018763761197295915697855647, 9.633929904774061610528280014358, 10.353630414221284935316307295490, 11.70474937526733155377139309031, 12.12449169060772942517225583493, 13.41825693296510597702553854419, 14.3410539945355273479340675987, 14.843707723023554238336502639075, 16.28114539832967930023142492879, 17.50390907852343062580579668944, 18.54583836464970069016265135266, 19.07592245543681270107542496355, 19.90210643706382878491348096933, 21.406311165406616323309941081204, 21.60464491442570165327265836531, 22.63433715554914563227641854400, 23.16849372447738363564890277227, 24.73113930292178258867000009399, 25.21526362124552340806882141044

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