↑

Properties

Label 1-1792-1792.1437-r0-0-0
Degree $1$
Conductor $1792$
Sign $-0.0388 + 0.999i$
Analytic cond. $8.32201$
Root an. cond. $8.32201$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.812 + 0.582i)3-s + (0.849 + 0.528i)5-s + (0.321 + 0.946i)9-s + (0.352 + 0.935i)11-s + (0.471 − 0.881i)13-s + (0.382 + 0.923i)15-s + (0.608 + 0.793i)17-s + (−0.683 + 0.729i)19-s + (−0.896 − 0.442i)23-s + (0.442 + 0.896i)25-s + (−0.290 + 0.956i)27-s + (0.773 − 0.634i)29-s + (−0.258 − 0.965i)31-s + (−0.258 + 0.965i)33-s + (0.227 + 0.973i)37-s + ⋯
L(s)  = 1  + (0.812 + 0.582i)3-s + (0.849 + 0.528i)5-s + (0.321 + 0.946i)9-s + (0.352 + 0.935i)11-s + (0.471 − 0.881i)13-s + (0.382 + 0.923i)15-s + (0.608 + 0.793i)17-s + (−0.683 + 0.729i)19-s + (−0.896 − 0.442i)23-s + (0.442 + 0.896i)25-s + (−0.290 + 0.956i)27-s + (0.773 − 0.634i)29-s + (−0.258 − 0.965i)31-s + (−0.258 + 0.965i)33-s + (0.227 + 0.973i)37-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $-0.0388 + 0.999i$
Analytic conductor: \(8.32201\)
Root analytic conductor: \(8.32201\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (1437, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (0:\ ),\ -0.0388 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.865550275 + 1.939436243i\)
\(L(\frac12)\) \(\approx\) \(1.865550275 + 1.939436243i\)
\(L(1)\) \(\approx\) \(1.525049223 + 0.6755326081i\)
\(L(1)\) \(\approx\) \(1.525049223 + 0.6755326081i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.812 + 0.582i)T \)
5 \( 1 + (0.849 + 0.528i)T \)
11 \( 1 + (0.352 + 0.935i)T \)
13 \( 1 + (0.471 - 0.881i)T \)
17 \( 1 + (0.608 + 0.793i)T \)
19 \( 1 + (-0.683 + 0.729i)T \)
23 \( 1 + (-0.896 - 0.442i)T \)
29 \( 1 + (0.773 - 0.634i)T \)
31 \( 1 + (-0.258 - 0.965i)T \)
37 \( 1 + (0.227 + 0.973i)T \)
41 \( 1 + (0.555 + 0.831i)T \)
43 \( 1 + (-0.0980 - 0.995i)T \)
47 \( 1 + (0.130 + 0.991i)T \)
53 \( 1 + (-0.935 + 0.352i)T \)
59 \( 1 + (-0.999 - 0.0327i)T \)
61 \( 1 + (-0.412 - 0.910i)T \)
67 \( 1 + (0.582 - 0.812i)T \)
71 \( 1 + (0.980 - 0.195i)T \)
73 \( 1 + (0.751 + 0.659i)T \)
79 \( 1 + (-0.793 - 0.608i)T \)
83 \( 1 + (-0.956 + 0.290i)T \)
89 \( 1 + (0.0654 - 0.997i)T \)
97 \( 1 + (0.707 - 0.707i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.83135149640497873815510061705, −19.46458718918578622350083697067, −18.45658682659303669709991687113, −18.01777585123697238047850810983, −17.12877453591978701421588377465, −16.28488843862134072870504154248, −15.73200285523546777911900921571, −14.28528116235386295663976787829, −14.19001161740243308513389444472, −13.471218893492765561110064831542, −12.692797077732515417268360851114, −11.97551503883191417261874631819, −11.06498379246720127871726599578, −10.01587201229033597587439726273, −9.10956317351109665689828127423, −8.85241253874715049532994005, −7.96906040048289991730023809040, −6.91244271826319356838674264760, −6.30761185740507792399921692801, −5.45231510730867708165986345826, −4.36478318007972071835919687384, −3.43740394019585800982480905258, −2.524999808438193554061550589239, −1.659635147986067077274384746316, −0.86018744076153301409421851027, 1.50378403207466849898428258753, 2.2154972908472304201927105308, 3.10190251922460187736765911004, 3.947157155491058290370000656365, 4.753161599695241041462627993182, 5.89740718574670797991154894855, 6.41964201677986386353352930182, 7.732646968459769122712929023811, 8.14403642152769774611870306723, 9.21117589033661224155353461385, 10.00642858043720498390442345426, 10.27077583373225125195136611774, 11.145250360289825086573769530324, 12.45116416148430652476057110720, 12.98847759345153448532334624916, 13.97837787307673645225591509382, 14.43280565643168758767455230295, 15.1688418694324515038685183761, 15.6917586817269287317703616897, 16.92280200885138133753653004741, 17.26471576704054040567030295290, 18.3903508609688063130067445077, 18.80996239392835809842392134470, 19.87832197777719788854154495486, 20.39379856337141499446982754165

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