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Properties

Label 1-311-311.49-r0-0-0
Degree $1$
Conductor $311$
Sign $0.943 - 0.330i$
Analytic cond. $1.44427$
Root an. cond. $1.44427$
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  + (0.688 − 0.724i)2-s + (0.688 + 0.724i)3-s + (−0.0506 − 0.998i)4-s + (0.979 + 0.201i)5-s + 6-s + (−0.954 + 0.299i)7-s + (−0.758 − 0.651i)8-s + (−0.0506 + 0.998i)9-s + (0.820 − 0.571i)10-s + (0.918 − 0.394i)11-s + (0.688 − 0.724i)12-s + (−0.0506 + 0.998i)13-s + (−0.440 + 0.897i)14-s + (0.528 + 0.848i)15-s + (−0.994 + 0.101i)16-s + (0.820 − 0.571i)17-s + ⋯
L(s)  = 1  + (0.688 − 0.724i)2-s + (0.688 + 0.724i)3-s + (−0.0506 − 0.998i)4-s + (0.979 + 0.201i)5-s + 6-s + (−0.954 + 0.299i)7-s + (−0.758 − 0.651i)8-s + (−0.0506 + 0.998i)9-s + (0.820 − 0.571i)10-s + (0.918 − 0.394i)11-s + (0.688 − 0.724i)12-s + (−0.0506 + 0.998i)13-s + (−0.440 + 0.897i)14-s + (0.528 + 0.848i)15-s + (−0.994 + 0.101i)16-s + (0.820 − 0.571i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.943 - 0.330i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.943 - 0.330i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(311\)
Sign: $0.943 - 0.330i$
Analytic conductor: \(1.44427\)
Root analytic conductor: \(1.44427\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{311} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 311,\ (0:\ ),\ 0.943 - 0.330i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.392610030 - 0.4066705588i\)
\(L(\frac12)\) \(\approx\) \(2.392610030 - 0.4066705588i\)
\(L(1)\) \(\approx\) \(1.899530200 - 0.3048976812i\)
\(L(1)\) \(\approx\) \(1.899530200 - 0.3048976812i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad311 \( 1 \)
good2 \( 1 + (0.688 - 0.724i)T \)
3 \( 1 + (0.688 + 0.724i)T \)
5 \( 1 + (0.979 + 0.201i)T \)
7 \( 1 + (-0.954 + 0.299i)T \)
11 \( 1 + (0.918 - 0.394i)T \)
13 \( 1 + (-0.0506 + 0.998i)T \)
17 \( 1 + (0.820 - 0.571i)T \)
19 \( 1 + (0.979 - 0.201i)T \)
23 \( 1 + (-0.758 - 0.651i)T \)
29 \( 1 + (-0.440 - 0.897i)T \)
31 \( 1 + (0.820 + 0.571i)T \)
37 \( 1 + (-0.954 - 0.299i)T \)
41 \( 1 + (-0.250 + 0.968i)T \)
43 \( 1 + (-0.954 + 0.299i)T \)
47 \( 1 + (-0.440 - 0.897i)T \)
53 \( 1 + (-0.954 + 0.299i)T \)
59 \( 1 + (-0.954 + 0.299i)T \)
61 \( 1 + (0.979 - 0.201i)T \)
67 \( 1 + (-0.250 - 0.968i)T \)
71 \( 1 + (-0.612 - 0.790i)T \)
73 \( 1 + (0.151 + 0.988i)T \)
79 \( 1 + (-0.250 + 0.968i)T \)
83 \( 1 + (0.528 - 0.848i)T \)
89 \( 1 + (-0.954 - 0.299i)T \)
97 \( 1 + (-0.612 - 0.790i)T \)
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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

−25.32620978866355118509775933844, −24.56861132719402193444862487616, −23.672340541974390565658308057535, −22.591883586235236761772695304590, −22.036078418836975977456273626567, −20.699304340633453245837881015926, −20.164378793167612592640634231435, −18.97175998892850993685584391532, −17.75973578260066848349782693497, −17.22813184887859218067752999864, −16.137944413505120752147405532812, −15.02295870811482813720252548695, −14.13247552306852659029572191617, −13.509975505396637287538457359576, −12.67435502510547083572440451599, −12.0410980217833658050662100455, −10.02425641484582041003800937303, −9.24652527481366670040686041609, −8.101496565096548271705481889289, −7.10627219471728460807202553488, −6.262060498066924324139263866571, −5.43691945178472106135349889060, −3.732326802764554242089589844935, −2.98694182084868730002948682666, −1.50313917640966670817692742468, 1.6626807237752285316071007657, 2.81867308647941495627484709394, 3.54740709474493115388825947863, 4.776636004501681312420196264697, 5.87959079459307172490241335014, 6.794110924206004131643011164683, 8.782245489112786783569811438741, 9.73197890963079659369887950488, 9.904693233622172685129603360833, 11.32385129428208196318428562908, 12.28155531804927577156435716634, 13.67414313341099444466341958521, 13.91030004436562506052107256914, 14.83067027304614881583922517925, 15.99397204297270871486343269129, 16.75894714870818337271671086377, 18.40094660379128541182579262976, 19.13906208248479495262800680339, 19.9596951450824045098382569330, 20.90476816216399805316955397722, 21.655940715950589230226883018287, 22.217835659796283186592181621, 22.94380422066010950885111407301, 24.58319758004564103792340982744, 25.00162989171474518416523759598

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