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Properties

Label 1-1840-1840.349-r0-0-0
Degree $1$
Conductor $1840$
Sign $-0.974 - 0.222i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.974 - 0.222i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.974 - 0.222i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $-0.974 - 0.222i$
Analytic conductor: \(8.54492\)
Root analytic conductor: \(8.54492\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1840} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1840,\ (0:\ ),\ -0.974 - 0.222i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02159720134 - 0.1913017024i\)
\(L(\frac12)\) \(\approx\) \(0.02159720134 - 0.1913017024i\)
\(L(1)\) \(\approx\) \(0.6354897185 - 0.1034372010i\)
\(L(1)\) \(\approx\) \(0.6354897185 - 0.1034372010i\)

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

−20.31675955528404367941433342644, −20.06305048645137542653835031316, −18.925491030353677445371696401299, −18.31476714992865833153885509022, −17.28789658485620547719530998056, −16.836738532066162828904908556932, −15.98634211412053731347000810725, −15.55766984112551478826447580341, −14.784791371052301412858787766857, −13.79974122869409466174953868293, −12.927196411639582546729175723420, −12.41677332663819217276574147841, −11.2917261654316818796820241976, −10.74982700934292627077766320916, −9.8998708549682930773030316786, −9.56858963406488990510192062449, −8.37843044516932095558123392833, −7.65110628190131861195981167712, −6.5083183346286564140898880396, −5.77792443303925850232866274537, −5.247849602799715713556480846028, −4.1068742779004323404003412677, −3.360835104128262774400121577, −2.721425903021493201812997987079, −1.0025962989231686310121303425, 0.08654547385558436348885100220, 1.39152493771504031877631320238, 2.43868569981406448780569126374, 3.0658493084354469338281160810, 4.47212720922227740191918099546, 5.25876199791499023483575054310, 6.025497818537790000640160210944, 6.9825909927844099382624633637, 7.302479233991993538523103591848, 8.308002420534597693697290229446, 9.44567096841626269549630914073, 9.873503094970381825209587396327, 10.988415727112110745234686476773, 11.80840887079881154869412434156, 12.263516843287417519582736945965, 13.16103304981495108639809124672, 13.613330903570252553993805463310, 14.51963626864866820556336813249, 15.590375508653129830044143807304, 16.25125205200449086744712367526, 16.78453252209814939478832095456, 17.8261203350605283380572531129, 18.31145303294562200092726372963, 18.98911197728344281800420729196, 19.61955255793839569802556158445

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