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Properties

Label 1-1792-1792.1405-r1-0-0
Degree $1$
Conductor $1792$
Sign $-0.624 - 0.780i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.624 - 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.624 - 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $-0.624 - 0.780i$
Analytic conductor: \(192.577\)
Root analytic conductor: \(192.577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (1405, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (1:\ ),\ -0.624 - 0.780i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2083630246 + 0.4336096477i\)
\(L(\frac12)\) \(\approx\) \(-0.2083630246 + 0.4336096477i\)
\(L(1)\) \(\approx\) \(0.6651424808 + 0.2886099489i\)
\(L(1)\) \(\approx\) \(0.6651424808 + 0.2886099489i\)

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

−19.42293309097506237022114350776, −18.68775801087199385568308484484, −18.08759587769522149677893108216, −17.18383729120646494546913310783, −16.50698628420384944544985284917, −16.27018633171074195948430431404, −15.361462215918138730476305813116, −14.118794338993823619969935546654, −13.48332153021328089891128327783, −12.79659140552410070792773439578, −11.951722341872025704216908801871, −11.50398289471972619249140139486, −10.59063007611701200859145278158, −9.73318731868717406636427075428, −9.00205013752699304730907067692, −8.12428176115255297107652833897, −7.19604646320496803547808650643, −6.34843268021015561622977043676, −5.58796946598073385085234450311, −4.84183000706084883512827260365, −4.28849166971158964356355348223, −2.89606145962083984837201566347, −1.78069433003935696475989536355, −0.79734689271190197223581367991, −0.12974596342459135015382048238, 1.27370817768138970924539941689, 2.22252694091847576777430980502, 3.35597973639581574936606453605, 4.1400177747851213249303000861, 5.27739924432428707438363759764, 5.821828577084420014242958798220, 6.59126362252331650589845358985, 7.51902823254158499137505616649, 8.00793788933766191551532976792, 9.68678547520374327914661850840, 10.02804408700171578996885111806, 10.64796449337038705369927304456, 11.42767157533787115522305648249, 12.36522610878621720272841351539, 12.75824733802001341650427258004, 13.884370483663841729177716932867, 14.72832017245666693454836105260, 15.39132810581796508559403914682, 15.948484427927475306137864702963, 17.10433407507253403535945842330, 17.54137217623831981935030984872, 18.13379039614649249852244072709, 18.85750095516543549851778283935, 19.62920485776071845988825610250, 20.70458786981238542240127783565

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