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Properties

Label 1-1792-1792.1005-r0-0-0
Degree $1$
Conductor $1792$
Sign $-0.480 + 0.877i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.0327 + 0.999i)3-s + (−0.352 − 0.935i)5-s + (−0.997 + 0.0654i)9-s + (0.227 + 0.973i)11-s + (0.773 − 0.634i)13-s + (0.923 − 0.382i)15-s + (−0.130 − 0.991i)17-s + (−0.812 − 0.582i)19-s + (−0.659 + 0.751i)23-s + (−0.751 + 0.659i)25-s + (−0.0980 − 0.995i)27-s + (0.290 + 0.956i)29-s + (−0.965 − 0.258i)31-s + (−0.965 + 0.258i)33-s + (−0.412 + 0.910i)37-s + ⋯
L(s)  = 1  + (0.0327 + 0.999i)3-s + (−0.352 − 0.935i)5-s + (−0.997 + 0.0654i)9-s + (0.227 + 0.973i)11-s + (0.773 − 0.634i)13-s + (0.923 − 0.382i)15-s + (−0.130 − 0.991i)17-s + (−0.812 − 0.582i)19-s + (−0.659 + 0.751i)23-s + (−0.751 + 0.659i)25-s + (−0.0980 − 0.995i)27-s + (0.290 + 0.956i)29-s + (−0.965 − 0.258i)31-s + (−0.965 + 0.258i)33-s + (−0.412 + 0.910i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.480 + 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.480 + 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $-0.480 + 0.877i$
Analytic conductor: \(8.32201\)
Root analytic conductor: \(8.32201\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (1005, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (0:\ ),\ -0.480 + 0.877i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4641501328 + 0.7834708702i\)
\(L(\frac12)\) \(\approx\) \(0.4641501328 + 0.7834708702i\)
\(L(1)\) \(\approx\) \(0.8505014733 + 0.2454623327i\)
\(L(1)\) \(\approx\) \(0.8505014733 + 0.2454623327i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.0327 + 0.999i)T \)
5 \( 1 + (-0.352 - 0.935i)T \)
11 \( 1 + (0.227 + 0.973i)T \)
13 \( 1 + (0.773 - 0.634i)T \)
17 \( 1 + (-0.130 - 0.991i)T \)
19 \( 1 + (-0.812 - 0.582i)T \)
23 \( 1 + (-0.659 + 0.751i)T \)
29 \( 1 + (0.290 + 0.956i)T \)
31 \( 1 + (-0.965 - 0.258i)T \)
37 \( 1 + (-0.412 + 0.910i)T \)
41 \( 1 + (-0.195 + 0.980i)T \)
43 \( 1 + (0.881 - 0.471i)T \)
47 \( 1 + (0.608 + 0.793i)T \)
53 \( 1 + (-0.973 + 0.227i)T \)
59 \( 1 + (0.162 - 0.986i)T \)
61 \( 1 + (0.528 + 0.849i)T \)
67 \( 1 + (0.999 - 0.0327i)T \)
71 \( 1 + (-0.555 + 0.831i)T \)
73 \( 1 + (0.896 + 0.442i)T \)
79 \( 1 + (0.991 + 0.130i)T \)
83 \( 1 + (0.995 + 0.0980i)T \)
89 \( 1 + (-0.321 + 0.946i)T \)
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

−19.54640836449089224977940236921, −19.188941393118191730976361987791, −18.62290734947886927875488798137, −17.962481624754636625289513872004, −17.11211648070633944960884917293, −16.36163479487092989167131383954, −15.48263147126705796344713753992, −14.45144195106718519279037874574, −14.160337029537265070439045848895, −13.331800968414609356576648097198, −12.44753465550774857943317582023, −11.78781374740596910995392125473, −10.88816157593076359395955917549, −10.59566785436764829066480668026, −9.13561602731611015727967643720, −8.36629035222256273707610679232, −7.84105916404408790908563454343, −6.779900740667648493363673422590, −6.27322758398576903080711790345, −5.70570603930069314679646087193, −4.01838962205612021841467610851, −3.56698122504449031139180776108, −2.38438410151655328159091936112, −1.73228951444253292945771931757, −0.34935138353467498875069285534, 1.08163548978085534822435094332, 2.317983696913737344643688490424, 3.44307352133218219036644931181, 4.16502303352906544994525829282, 4.915842944130376893437046707368, 5.49056691665591364395900230522, 6.60269469102962773616160873390, 7.69843970160617024107903442442, 8.45097262753599862992521437551, 9.22933745824322509200861046321, 9.7027182011140547091288480151, 10.729595972397074331797662438730, 11.37776231663967734668567495716, 12.21566517558837660589562892186, 12.95631240481769894418475323848, 13.80570513951227981777746996981, 14.69892293636890521686637816090, 15.56290381444733227991916506185, 15.833300732087409156836182190048, 16.6707818000766638068749926612, 17.42429528439351175437558553135, 17.99963592526188956782090040009, 19.19751035193598935727040953661, 20.12845541941165915197296991455, 20.3388714150891237343324198725

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