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Properties

Label 1-1840-1840.163-r0-0-0
Degree $1$
Conductor $1840$
Sign $-0.164 + 0.986i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.164 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.164 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $-0.164 + 0.986i$
Analytic conductor: \(8.54492\)
Root analytic conductor: \(8.54492\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1840} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1840,\ (0:\ ),\ -0.164 + 0.986i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3951485104 + 0.4664853615i\)
\(L(\frac12)\) \(\approx\) \(0.3951485104 + 0.4664853615i\)
\(L(1)\) \(\approx\) \(0.6612877946 + 0.1018865774i\)
\(L(1)\) \(\approx\) \(0.6612877946 + 0.1018865774i\)

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

−19.67729144486457513243404262718, −19.08329165116965796002982753513, −18.64148116263821597135616519070, −17.601873411349837131818478013230, −16.92134123172349392056886909007, −16.42249431497364268567683381665, −15.79796086581123075428799123574, −14.83105942220725894607489089326, −13.69629973722706572917946227521, −13.37166986141189850429255342970, −12.31031558188393782564199384326, −11.808404304546929685251944574535, −11.07845682878779884172616005098, −10.24672210598065711670196059940, −9.4447468697089705362152614846, −8.742284728440703880988547389667, −7.53600941760269566232612749976, −6.59945369690593618423938223414, −6.41235784722988901595372361262, −5.46644553449612088946507433917, −4.29968425420304135767420154436, −3.829091813189345079947504959486, −2.46408019038564546405684375620, −1.48606148608113780493270571290, −0.315671086201549161569450477731, 0.896516113991602564538691265223, 2.10042050391170985888910213199, 3.34983727837019580239361268120, 4.04775539103983255311598542285, 4.96550728084468008885882362916, 5.8319268081569100751083764906, 6.61744747010564755017770800452, 6.981224627635671148896549962510, 8.36919609102737796556765784774, 9.184797425595445475181907955020, 9.93723230687208829755947156558, 10.629906806788367791070292329278, 11.30373026536172517214713076387, 12.31067952840071292827519898754, 12.71604257640641602673348537891, 13.44415816057919753136508486609, 14.646029238860831081160523122523, 15.53386529111707414332070552349, 15.73034096918294839175749636939, 16.90603389259867004772583569964, 17.27787409716501180902674973910, 17.94044834284231484436534433615, 18.93405756586600066487201981594, 19.55015984658662962592440896999, 20.3202387826839821910080980441

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