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Properties

Label 1-1840-1840.1083-r0-0-0
Degree $1$
Conductor $1840$
Sign $0.581 + 0.813i$
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  + (0.959 − 0.281i)3-s + (−0.989 + 0.142i)7-s + (0.841 − 0.540i)9-s + (−0.909 − 0.415i)11-s + (−0.142 + 0.989i)13-s + (−0.755 + 0.654i)17-s + (0.755 + 0.654i)19-s + (−0.909 + 0.415i)21-s + (0.654 − 0.755i)27-s + (0.755 − 0.654i)29-s + (0.959 + 0.281i)31-s + (−0.989 − 0.142i)33-s + (0.841 − 0.540i)37-s + (0.142 + 0.989i)39-s + (−0.841 − 0.540i)41-s + ⋯
L(s)  = 1  + (0.959 − 0.281i)3-s + (−0.989 + 0.142i)7-s + (0.841 − 0.540i)9-s + (−0.909 − 0.415i)11-s + (−0.142 + 0.989i)13-s + (−0.755 + 0.654i)17-s + (0.755 + 0.654i)19-s + (−0.909 + 0.415i)21-s + (0.654 − 0.755i)27-s + (0.755 − 0.654i)29-s + (0.959 + 0.281i)31-s + (−0.989 − 0.142i)33-s + (0.841 − 0.540i)37-s + (0.142 + 0.989i)39-s + (−0.841 − 0.540i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.581 + 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.581 + 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $0.581 + 0.813i$
Analytic conductor: \(8.54492\)
Root analytic conductor: \(8.54492\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1840} (1083, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1840,\ (0:\ ),\ 0.581 + 0.813i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.454411212 + 0.7485208555i\)
\(L(\frac12)\) \(\approx\) \(1.454411212 + 0.7485208555i\)
\(L(1)\) \(\approx\) \(1.220749604 + 0.07551861576i\)
\(L(1)\) \(\approx\) \(1.220749604 + 0.07551861576i\)

Euler product

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

−19.9986418148155558190475267840, −19.64078359144589758518627152935, −18.43167768772780881645656942348, −18.17635387353726809521916474279, −16.996800630955849267197653453725, −16.1065967116667758970781010777, −15.45916584694821397065440006338, −15.18967204789359201956234924544, −13.94528531523343263042614505801, −13.356339321749920013810983938509, −12.9216868389147104630569685366, −11.95753386325883113671725976570, −10.80538014480257564139297011863, −10.02930117636341027981612987401, −9.65258148544980076313387306007, −8.69019290583932664742579383159, −7.93704927145561072540655540497, −7.17533186479310560411529629728, −6.43222459839661922794001702204, −5.10416447711271850756175936959, −4.62248168608859782341765072937, −3.21508535958845441764980168663, −3.02613427429516394943349551863, −2.02907689099955110327241372248, −0.52992712861325864696979295807, 1.12159733060132356350221792552, 2.29189806264516693345222604749, 2.88695152665576402389207706845, 3.78063895764539361377877452707, 4.58097917326572996978140659451, 5.87450634087522880955239698912, 6.56444414972283713007707461403, 7.35376309266831835323385458032, 8.243184846903025183941646863681, 8.83800776010740091522684312071, 9.749094399523650185932889988308, 10.19837466495254894839290303523, 11.35879560599145444164595392364, 12.31073560991345503562854044403, 12.90141102521650820275411295863, 13.71863227784021776348009265827, 14.07929997786549151179809419024, 15.24147658647674443542139079194, 15.7177039148567237384327883513, 16.399813833252680428331551287212, 17.348729574007895566829023824424, 18.48647009830988141624285769441, 18.71358895478059378874713050439, 19.61914390175227933483802468817, 19.99627112043748662363647194535

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