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Properties

Label 1-1792-1792.387-r1-0-0
Degree $1$
Conductor $1792$
Sign $-0.999 - 0.0388i$
Analytic cond. $192.577$
Root an. cond. $192.577$
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  + (−0.986 + 0.162i)3-s + (−0.227 − 0.973i)5-s + (0.946 − 0.321i)9-s + (0.412 − 0.910i)11-s + (0.290 + 0.956i)13-s + (0.382 + 0.923i)15-s + (−0.608 − 0.793i)17-s + (0.0327 + 0.999i)19-s + (0.442 − 0.896i)23-s + (−0.896 + 0.442i)25-s + (−0.881 + 0.471i)27-s + (0.0980 − 0.995i)29-s + (0.258 + 0.965i)31-s + (−0.258 + 0.965i)33-s + (0.528 − 0.849i)37-s + ⋯
L(s)  = 1  + (−0.986 + 0.162i)3-s + (−0.227 − 0.973i)5-s + (0.946 − 0.321i)9-s + (0.412 − 0.910i)11-s + (0.290 + 0.956i)13-s + (0.382 + 0.923i)15-s + (−0.608 − 0.793i)17-s + (0.0327 + 0.999i)19-s + (0.442 − 0.896i)23-s + (−0.896 + 0.442i)25-s + (−0.881 + 0.471i)27-s + (0.0980 − 0.995i)29-s + (0.258 + 0.965i)31-s + (−0.258 + 0.965i)33-s + (0.528 − 0.849i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 - 0.0388i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 - 0.0388i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $-0.999 - 0.0388i$
Analytic conductor: \(192.577\)
Root analytic conductor: \(192.577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (387, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (1:\ ),\ -0.999 - 0.0388i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01214362257 - 0.6253734181i\)
\(L(\frac12)\) \(\approx\) \(0.01214362257 - 0.6253734181i\)
\(L(1)\) \(\approx\) \(0.7060419785 - 0.1835411524i\)
\(L(1)\) \(\approx\) \(0.7060419785 - 0.1835411524i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.986 + 0.162i)T \)
5 \( 1 + (-0.227 - 0.973i)T \)
11 \( 1 + (0.412 - 0.910i)T \)
13 \( 1 + (0.290 + 0.956i)T \)
17 \( 1 + (-0.608 - 0.793i)T \)
19 \( 1 + (0.0327 + 0.999i)T \)
23 \( 1 + (0.442 - 0.896i)T \)
29 \( 1 + (0.0980 - 0.995i)T \)
31 \( 1 + (0.258 + 0.965i)T \)
37 \( 1 + (0.528 - 0.849i)T \)
41 \( 1 + (0.831 - 0.555i)T \)
43 \( 1 + (-0.634 + 0.773i)T \)
47 \( 1 + (0.130 + 0.991i)T \)
53 \( 1 + (-0.910 - 0.412i)T \)
59 \( 1 + (-0.683 - 0.729i)T \)
61 \( 1 + (-0.935 - 0.352i)T \)
67 \( 1 + (0.162 + 0.986i)T \)
71 \( 1 + (-0.195 - 0.980i)T \)
73 \( 1 + (0.659 - 0.751i)T \)
79 \( 1 + (-0.793 - 0.608i)T \)
83 \( 1 + (-0.471 + 0.881i)T \)
89 \( 1 + (0.997 + 0.0654i)T \)
97 \( 1 + (0.707 - 0.707i)T \)
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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

−20.114786145030762895679271590691, −19.726542554097492446125011963270, −18.654166482125915739150397463326, −18.17451785376715801357245664095, −17.39228015547532977481304214168, −17.06202970072465462980702655241, −15.717837487615997076605550439347, −15.35447347685503648036825000941, −14.70518736732486281315961503667, −13.45943538228631155682284599353, −12.96559052354161609099487671305, −12.02615162432504527361671401356, −11.34232857512962859897347206345, −10.72450866162167343346344408609, −10.10055003108244296568389883583, −9.19985187883321678971596073529, −7.951502216340023542780642457649, −7.24130608012440475868559535804, −6.596253646647971362345485419386, −5.9067399482017889939265188498, −4.92545445889147711179777755905, −4.11904782391761366658265670556, −3.13344164484084003479557719468, −2.07347084326586577185044966067, −1.04240107701392927336263812271, 0.16927865322343487433073636008, 0.94764115315509113077484946776, 1.85145071588033571280801474712, 3.351353415178205911414586240532, 4.38020737186124259160887603799, 4.72169644711488227284371248420, 5.88333279017867481208681073257, 6.324931698375812907140438804797, 7.3541233213200195200409852862, 8.34891183718299211616328242945, 9.129909591236249139089902950681, 9.74294921163870266291462467289, 10.932303405167587525728101175270, 11.37261557276960416685710193991, 12.16400157209819967883324496179, 12.72309739815263292135805615954, 13.672495751653779823542672721919, 14.36598019695234807006271352313, 15.59340115876992150180103337044, 16.22451567531946823854848395745, 16.55770436740581628884221340638, 17.289951295781968690153069235913, 18.119825515497344381239513311078, 18.928060946382552263382724186074, 19.52771337509051583957314693431

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