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Properties

Label 1-1840-1840.1189-r0-0-0
Degree $1$
Conductor $1840$
Sign $0.258 + 0.965i$
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  + (0.909 + 0.415i)3-s + (0.841 − 0.540i)7-s + (0.654 + 0.755i)9-s + (−0.989 − 0.142i)11-s + (−0.540 + 0.841i)13-s + (0.959 + 0.281i)17-s + (0.281 + 0.959i)19-s + (0.989 − 0.142i)21-s + (0.281 + 0.959i)27-s + (−0.281 + 0.959i)29-s + (0.415 + 0.909i)31-s + (−0.841 − 0.540i)33-s + (−0.755 + 0.654i)37-s + (−0.841 + 0.540i)39-s + (0.654 − 0.755i)41-s + ⋯
L(s)  = 1  + (0.909 + 0.415i)3-s + (0.841 − 0.540i)7-s + (0.654 + 0.755i)9-s + (−0.989 − 0.142i)11-s + (−0.540 + 0.841i)13-s + (0.959 + 0.281i)17-s + (0.281 + 0.959i)19-s + (0.989 − 0.142i)21-s + (0.281 + 0.959i)27-s + (−0.281 + 0.959i)29-s + (0.415 + 0.909i)31-s + (−0.841 − 0.540i)33-s + (−0.755 + 0.654i)37-s + (−0.841 + 0.540i)39-s + (0.654 − 0.755i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $0.258 + 0.965i$
Analytic conductor: \(8.54492\)
Root analytic conductor: \(8.54492\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1840} (1189, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1840,\ (0:\ ),\ 0.258 + 0.965i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.807366064 + 1.387018134i\)
\(L(\frac12)\) \(\approx\) \(1.807366064 + 1.387018134i\)
\(L(1)\) \(\approx\) \(1.440667584 + 0.3948464877i\)
\(L(1)\) \(\approx\) \(1.440667584 + 0.3948464877i\)

Euler product

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

−20.04335526118007344147138537103, −19.20489484729510226012119593216, −18.56146545440857773448318952197, −17.89668638823713527991665394073, −17.36045386058502479673917635467, −16.10206228783290733395865408582, −15.29517331569577951339551355784, −14.93164654063903508134802601744, −14.0980801296104610098444839467, −13.31920529709995957974482567098, −12.68011466059461619500277313090, −11.91104091962227625118688118342, −11.078357367242454237616383665191, −10.0105937388481344253486095948, −9.46196511353835421605876239825, −8.439987601785836306991266160694, −7.79373496461069439621544657531, −7.45470621100107137493841179145, −6.190624222685872036663686085722, −5.24491103170785001202150702227, −4.59270322526696234896223041574, −3.29340365425142866706001844460, −2.63130702664332845619091059802, −1.91739552655452701029744251549, −0.70008291793839935360345825192, 1.36990500100332399144527320649, 2.07350141421798303763449085892, 3.191680146932810919972161354, 3.84322066302149821732549524860, 4.896216906823166222519278886825, 5.31699471769638950933849117771, 6.80151127261732751141041115739, 7.58887907392748170555725528092, 8.15960056386039326529077190964, 8.84455119044799799138007480575, 9.99860569142255362130664580979, 10.276481078230706774915082662241, 11.20996511572653242331213903784, 12.16906864129029265052917932541, 12.98726627594491230804548271396, 13.963522486397788648160845602107, 14.3036077447622036257819229209, 14.95050233430053511307742185186, 15.92837887449846021590170125612, 16.492840328626679548709382618356, 17.274431846110396952208885031783, 18.30390270485519955156567510166, 18.86522870421456454030553079, 19.61597733611218177681231687992, 20.451601815011638371460604868586

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