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Properties

Label 1-1840-1840.1349-r1-0-0
Degree $1$
Conductor $1840$
Sign $0.769 - 0.638i$
Analytic cond. $197.735$
Root an. cond. $197.735$
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  + (−0.755 − 0.654i)3-s + (−0.415 − 0.909i)7-s + (0.142 + 0.989i)9-s + (0.281 + 0.959i)11-s + (0.909 + 0.415i)13-s + (0.841 − 0.540i)17-s + (−0.540 + 0.841i)19-s + (−0.281 + 0.959i)21-s + (0.540 − 0.841i)27-s + (−0.540 − 0.841i)29-s + (−0.654 − 0.755i)31-s + (0.415 − 0.909i)33-s + (0.989 − 0.142i)37-s + (−0.415 − 0.909i)39-s + (0.142 − 0.989i)41-s + ⋯
L(s)  = 1  + (−0.755 − 0.654i)3-s + (−0.415 − 0.909i)7-s + (0.142 + 0.989i)9-s + (0.281 + 0.959i)11-s + (0.909 + 0.415i)13-s + (0.841 − 0.540i)17-s + (−0.540 + 0.841i)19-s + (−0.281 + 0.959i)21-s + (0.540 − 0.841i)27-s + (−0.540 − 0.841i)29-s + (−0.654 − 0.755i)31-s + (0.415 − 0.909i)33-s + (0.989 − 0.142i)37-s + (−0.415 − 0.909i)39-s + (0.142 − 0.989i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.769 - 0.638i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.769 - 0.638i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $0.769 - 0.638i$
Analytic conductor: \(197.735\)
Root analytic conductor: \(197.735\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1840} (1349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1840,\ (1:\ ),\ 0.769 - 0.638i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.359071813 - 0.4899513293i\)
\(L(\frac12)\) \(\approx\) \(1.359071813 - 0.4899513293i\)
\(L(1)\) \(\approx\) \(0.8293055055 - 0.1819194704i\)
\(L(1)\) \(\approx\) \(0.8293055055 - 0.1819194704i\)

Euler product

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

−20.01030757265494581682045434527, −19.29179310777989546714874406879, −18.34630813179776883372199468123, −18.01194426439259866119455792594, −16.86919433457594155702233979706, −16.38085501195474023789331741887, −15.77040488846017525053507793693, −14.981569647637573221670170659022, −14.37955866951733263561793132394, −13.01097426578102774127304175982, −12.74355932950828277300030823271, −11.54408004989770593246640794847, −11.22242254859716364483659029775, −10.3576790035611014427815504992, −9.509025472939293753957755438391, −8.80255671481464349327134883538, −8.13431518881444383292494048388, −6.70224567696057676987574228377, −6.11891321631019307037204036054, −5.51691219116116210723045292437, −4.690461722670838021054743077460, −3.459949379324539620021771310615, −3.15763407014298654723125499165, −1.59074043317200219862689603803, −0.54915371969194486498737779386, 0.532913134029825431254922372574, 1.42005942791875172674717349528, 2.25353265552293405352784495629, 3.696432230443545662870781312456, 4.27282540571602494353692585133, 5.37959570553619545138368375775, 6.16896340671373517020070867076, 6.893703484619808305225074227041, 7.524902478980981272186421688996, 8.28562139507274666589981071458, 9.60416953642300558509115175518, 10.09104758285136696329999665533, 11.05978685238198739513497622755, 11.61701548541944104213800992937, 12.5624806235436707851073846424, 13.04072304600242247430714447842, 13.87179154827029589093070261166, 14.53583926759906975475621695939, 15.6413410599275673574506745603, 16.50951062764827981168631732963, 16.8846380630491940050129783516, 17.6114199814368909571024562323, 18.53862270953826131699273755445, 18.90505309841009641576670534329, 19.85885346911715135861304949139

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