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Properties

Label 1-1792-1792.283-r0-0-0
Degree $1$
Conductor $1792$
Sign $0.996 + 0.0857i$
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  + (0.849 + 0.528i)3-s + (−0.986 + 0.162i)5-s + (0.442 + 0.896i)9-s + (−0.729 − 0.683i)11-s + (−0.773 − 0.634i)13-s + (−0.923 − 0.382i)15-s + (0.793 + 0.608i)17-s + (0.910 + 0.412i)19-s + (0.321 − 0.946i)23-s + (0.946 − 0.321i)25-s + (−0.0980 + 0.995i)27-s + (0.290 − 0.956i)29-s + (0.258 − 0.965i)31-s + (−0.258 − 0.965i)33-s + (−0.582 + 0.812i)37-s + ⋯
L(s)  = 1  + (0.849 + 0.528i)3-s + (−0.986 + 0.162i)5-s + (0.442 + 0.896i)9-s + (−0.729 − 0.683i)11-s + (−0.773 − 0.634i)13-s + (−0.923 − 0.382i)15-s + (0.793 + 0.608i)17-s + (0.910 + 0.412i)19-s + (0.321 − 0.946i)23-s + (0.946 − 0.321i)25-s + (−0.0980 + 0.995i)27-s + (0.290 − 0.956i)29-s + (0.258 − 0.965i)31-s + (−0.258 − 0.965i)33-s + (−0.582 + 0.812i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0857i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0857i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.996 + 0.0857i$
Analytic conductor: \(8.32201\)
Root analytic conductor: \(8.32201\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (283, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (0:\ ),\ 0.996 + 0.0857i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.650414385 + 0.07088282549i\)
\(L(\frac12)\) \(\approx\) \(1.650414385 + 0.07088282549i\)
\(L(1)\) \(\approx\) \(1.165084042 + 0.1391871995i\)
\(L(1)\) \(\approx\) \(1.165084042 + 0.1391871995i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.849 + 0.528i)T \)
5 \( 1 + (-0.986 + 0.162i)T \)
11 \( 1 + (-0.729 - 0.683i)T \)
13 \( 1 + (-0.773 - 0.634i)T \)
17 \( 1 + (0.793 + 0.608i)T \)
19 \( 1 + (0.910 + 0.412i)T \)
23 \( 1 + (0.321 - 0.946i)T \)
29 \( 1 + (0.290 - 0.956i)T \)
31 \( 1 + (0.258 - 0.965i)T \)
37 \( 1 + (-0.582 + 0.812i)T \)
41 \( 1 + (0.195 + 0.980i)T \)
43 \( 1 + (-0.881 - 0.471i)T \)
47 \( 1 + (-0.991 - 0.130i)T \)
53 \( 1 + (0.683 - 0.729i)T \)
59 \( 1 + (-0.935 - 0.352i)T \)
61 \( 1 + (0.999 + 0.0327i)T \)
67 \( 1 + (0.528 - 0.849i)T \)
71 \( 1 + (0.555 + 0.831i)T \)
73 \( 1 + (0.0654 - 0.997i)T \)
79 \( 1 + (0.608 + 0.793i)T \)
83 \( 1 + (0.995 - 0.0980i)T \)
89 \( 1 + (0.659 - 0.751i)T \)
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

−19.91830876679242009311795841347, −19.65006891763337678968183730311, −18.81391612140348476202329249502, −18.17430468223965122285217828615, −17.422823817178975239057444274690, −16.2462108838258088042376095253, −15.75155434900210693864315047677, −14.96157253107238405496907247428, −14.29822995132874114592869010018, −13.57734960909874810686628030052, −12.599793231040227431027832414663, −12.16178450014265084600879340664, −11.44752043328686997832055718500, −10.28396400546704735147247046287, −9.45875893974135419809004691748, −8.806249529081615138849169188661, −7.83807266507647566086785051217, −7.29621639261488821729522369704, −6.89161530766323502852658807775, −5.28606460880720934052694928883, −4.69461115735261406619581148111, −3.528083355286057514039558926893, −2.99106282943907680488653149837, −1.94185504527990025595062064233, −0.88669433650951589053295979643, 0.68785476945581601845512280745, 2.218067518699309315144520078789, 3.14775750172052369098000635942, 3.54258857263683439220052093105, 4.65532987441859757648691084673, 5.27766660238079996245072658109, 6.4907734170157684371195461837, 7.723208279482721255678698053183, 7.94506131653393109653523649029, 8.622448250473418122054859852785, 9.871568813020234276937365250668, 10.22735331599403438076843656063, 11.161612183616700721972725456, 11.97785415229589928645116487373, 12.8454292826267528922399219349, 13.59196806055438421763524031022, 14.52209971208569898256480830551, 15.033190570646488043334448176995, 15.66570993730516334566816903658, 16.39906375441069917543100220961, 17.00934458172952574044689047271, 18.37581955990554848952996052408, 18.84765788972357722666046720194, 19.52632057911231877584628170226, 20.20329097959631978040970701431

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