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Properties

Label 1-592-592.171-r0-0-0
Degree $1$
Conductor $592$
Sign $0.412 + 0.911i$
Analytic cond. $2.74923$
Root an. cond. $2.74923$
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  + (0.866 + 0.5i)3-s + (0.5 − 0.866i)5-s + (−0.5 + 0.866i)7-s + (0.5 + 0.866i)9-s − i·11-s + (−0.5 + 0.866i)13-s + (0.866 − 0.5i)15-s + (0.866 − 0.5i)17-s + (0.5 − 0.866i)19-s + (−0.866 + 0.5i)21-s + i·23-s + (−0.5 − 0.866i)25-s − i·27-s − 29-s − i·31-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)3-s + (0.5 − 0.866i)5-s + (−0.5 + 0.866i)7-s + (0.5 + 0.866i)9-s − i·11-s + (−0.5 + 0.866i)13-s + (0.866 − 0.5i)15-s + (0.866 − 0.5i)17-s + (0.5 − 0.866i)19-s + (−0.866 + 0.5i)21-s + i·23-s + (−0.5 − 0.866i)25-s − i·27-s − 29-s − i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.412 + 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.412 + 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(592\)    =    \(2^{4} \cdot 37\)
Sign: $0.412 + 0.911i$
Analytic conductor: \(2.74923\)
Root analytic conductor: \(2.74923\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{592} (171, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 592,\ (0:\ ),\ 0.412 + 0.911i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.597056917 + 1.030406861i\)
\(L(\frac12)\) \(\approx\) \(1.597056917 + 1.030406861i\)
\(L(1)\) \(\approx\) \(1.383674531 + 0.3924497782i\)
\(L(1)\) \(\approx\) \(1.383674531 + 0.3924497782i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
37 \( 1 \)
good3 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 - iT \)
13 \( 1 + (-0.5 + 0.866i)T \)
17 \( 1 + (0.866 - 0.5i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + iT \)
29 \( 1 - T \)
31 \( 1 - iT \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + T \)
47 \( 1 - T \)
53 \( 1 + (0.866 - 0.5i)T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (0.866 + 0.5i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + T \)
79 \( 1 + (0.866 + 0.5i)T \)
83 \( 1 + (0.866 - 0.5i)T \)
89 \( 1 + (0.866 - 0.5i)T \)
97 \( 1 + iT \)
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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

−22.850651165184957751421682745351, −22.32823554908802468706787567356, −21.14241234966679632093621875201, −20.51499434551204352404882768683, −19.54208916471601224076529396220, −18.8731853272649915120993069068, −18.27020513703956729891620116950, −17.170776438401607415179825706552, −16.39658945046940564853458256268, −15.10089167623847417871340551286, −14.46834320608463027479350355775, −13.73875973405241487812214996701, −13.07630940021126294397044546588, −12.12307007641638646355316799900, −10.73602483470161885215165854825, −10.13886032766993658748901809604, −9.28232075400864475937728613512, −7.98486899002719306293403225260, −7.47089812937698453047548122631, −6.41676905592287894046311478593, −5.65654284655781032097691040979, −3.79433141939891887518478602418, −3.26120728756539377651092000858, −2.2624365790731665134100959445, −0.89831059564374841427475953910, 1.63316240861120682215135899819, 2.46648601372276082742801153618, 3.58363236708180421594991123349, 4.848023550668761221532443234771, 5.32182792377962026940798816440, 6.80875380239946309784544621164, 7.79519771715745524920033056257, 8.91190949098003574732025954671, 9.55310753860394981042945316208, 9.85086537993248410133454006802, 11.491818348000980651847255382336, 12.41689706745566274022072103498, 13.15677959649340889578623940023, 14.063247839828663427329218795325, 14.897865511483856883873918584462, 15.79424273917903596374592742718, 16.38516586699144880742280932637, 17.3946851784297274044909765729, 18.39608104934523591691777688584, 19.368198649219262503163130850651, 20.01178420801969420252405605414, 20.84948281381959287972948641505, 21.56307512586640165684736687181, 22.13123071802998217807978326831, 23.30762466601651506907999217473

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