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Properties

Label 1-1840-1840.19-r0-0-0
Degree $1$
Conductor $1840$
Sign $-0.671 - 0.740i$
Analytic cond. $8.54492$
Root an. cond. $8.54492$
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  + (0.540 + 0.841i)3-s + (−0.959 + 0.281i)7-s + (−0.415 + 0.909i)9-s + (−0.755 + 0.654i)11-s + (−0.281 + 0.959i)13-s + (−0.142 + 0.989i)17-s + (0.989 − 0.142i)19-s + (−0.755 − 0.654i)21-s + (−0.989 + 0.142i)27-s + (−0.989 − 0.142i)29-s + (−0.841 − 0.540i)31-s + (−0.959 − 0.281i)33-s + (−0.909 − 0.415i)37-s + (−0.959 + 0.281i)39-s + (−0.415 − 0.909i)41-s + ⋯
L(s)  = 1  + (0.540 + 0.841i)3-s + (−0.959 + 0.281i)7-s + (−0.415 + 0.909i)9-s + (−0.755 + 0.654i)11-s + (−0.281 + 0.959i)13-s + (−0.142 + 0.989i)17-s + (0.989 − 0.142i)19-s + (−0.755 − 0.654i)21-s + (−0.989 + 0.142i)27-s + (−0.989 − 0.142i)29-s + (−0.841 − 0.540i)31-s + (−0.959 − 0.281i)33-s + (−0.909 − 0.415i)37-s + (−0.959 + 0.281i)39-s + (−0.415 − 0.909i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.671 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.671 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $-0.671 - 0.740i$
Analytic conductor: \(8.54492\)
Root analytic conductor: \(8.54492\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1840} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1840,\ (0:\ ),\ -0.671 - 0.740i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2113686243 + 0.4769072522i\)
\(L(\frac12)\) \(\approx\) \(-0.2113686243 + 0.4769072522i\)
\(L(1)\) \(\approx\) \(0.7561788713 + 0.4450337376i\)
\(L(1)\) \(\approx\) \(0.7561788713 + 0.4450337376i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
23 \( 1 \)
good3 \( 1 + (0.540 + 0.841i)T \)
7 \( 1 + (-0.959 + 0.281i)T \)
11 \( 1 + (-0.755 + 0.654i)T \)
13 \( 1 + (-0.281 + 0.959i)T \)
17 \( 1 + (-0.142 + 0.989i)T \)
19 \( 1 + (0.989 - 0.142i)T \)
29 \( 1 + (-0.989 - 0.142i)T \)
31 \( 1 + (-0.841 - 0.540i)T \)
37 \( 1 + (-0.909 - 0.415i)T \)
41 \( 1 + (-0.415 - 0.909i)T \)
43 \( 1 + (0.540 + 0.841i)T \)
47 \( 1 + T \)
53 \( 1 + (0.281 + 0.959i)T \)
59 \( 1 + (0.281 - 0.959i)T \)
61 \( 1 + (0.540 - 0.841i)T \)
67 \( 1 + (0.755 + 0.654i)T \)
71 \( 1 + (-0.654 + 0.755i)T \)
73 \( 1 + (-0.142 - 0.989i)T \)
79 \( 1 + (-0.959 - 0.281i)T \)
83 \( 1 + (-0.909 - 0.415i)T \)
89 \( 1 + (0.841 - 0.540i)T \)
97 \( 1 + (0.415 + 0.909i)T \)
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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

−19.73101789533570501182877023827, −18.85850263260285845747926637509, −18.3844355439305875186462513503, −17.69008950193643835693438868567, −16.705842068378112871709274765518, −15.96175523470532323554539626791, −15.303244798946252846345967172456, −14.32130678380646177714982089327, −13.57094605304998252914624189404, −13.146478752609066550728037227657, −12.41128423324577243810083180159, −11.623737721126433456948426296936, −10.61730662103778233352767578290, −9.809086527397966636742884667912, −9.05565774227892756133247573703, −8.233766509004691063554767579410, −7.32960793217579047423097579460, −7.00083012832995790458102548220, −5.78239670694361842405320357461, −5.29754902626544928507251888650, −3.726583404127210680826354652626, −3.08977095732559459694695108797, −2.478838687073085058805721100300, −1.13898480838342528428979741091, −0.17101302697229315926443545310, 1.870472658412954164526549422425, 2.56104782697889459696700558041, 3.56313042348075291456776195824, 4.15853108662615880169387356611, 5.20228877829442907694377471161, 5.86463575207900566465963666879, 7.05281318990330100735228376285, 7.68376046784280084466882120805, 8.80587395690884645678807029813, 9.337799228282279156828389796220, 9.98975860744118621917862870542, 10.69087596965885660096209017272, 11.59807054828761082374341207199, 12.581499680275412937494756661477, 13.19887665566775900123072058676, 14.06383994351085244707300737051, 14.7934976013457493508771216621, 15.58422786058270388655732153612, 15.98937188274961613239746421737, 16.80922154348093077857628187255, 17.52206198849851965897379083049, 18.78644224994005430404803836288, 19.03353721176098151737068047366, 20.08624880373906197460744529857, 20.45943600206456840530771283387

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