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Properties

Label 1-1792-1792.1371-r0-0-0
Degree $1$
Conductor $1792$
Sign $-0.534 - 0.844i$
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  + (0.471 + 0.881i)3-s + (−0.773 + 0.634i)5-s + (−0.555 + 0.831i)9-s + (0.290 + 0.956i)11-s + (−0.634 + 0.773i)13-s + (−0.923 − 0.382i)15-s + (−0.923 + 0.382i)17-s + (−0.995 + 0.0980i)19-s + (0.980 − 0.195i)23-s + (0.195 − 0.980i)25-s + (−0.995 − 0.0980i)27-s + (−0.956 − 0.290i)29-s + (0.707 + 0.707i)31-s + (−0.707 + 0.707i)33-s + (−0.0980 + 0.995i)37-s + ⋯
L(s)  = 1  + (0.471 + 0.881i)3-s + (−0.773 + 0.634i)5-s + (−0.555 + 0.831i)9-s + (0.290 + 0.956i)11-s + (−0.634 + 0.773i)13-s + (−0.923 − 0.382i)15-s + (−0.923 + 0.382i)17-s + (−0.995 + 0.0980i)19-s + (0.980 − 0.195i)23-s + (0.195 − 0.980i)25-s + (−0.995 − 0.0980i)27-s + (−0.956 − 0.290i)29-s + (0.707 + 0.707i)31-s + (−0.707 + 0.707i)33-s + (−0.0980 + 0.995i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.534 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.534 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $-0.534 - 0.844i$
Analytic conductor: \(8.32201\)
Root analytic conductor: \(8.32201\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (1371, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (0:\ ),\ -0.534 - 0.844i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2892571637 + 0.5255455810i\)
\(L(\frac12)\) \(\approx\) \(-0.2892571637 + 0.5255455810i\)
\(L(1)\) \(\approx\) \(0.6829516724 + 0.5100090882i\)
\(L(1)\) \(\approx\) \(0.6829516724 + 0.5100090882i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.471 + 0.881i)T \)
5 \( 1 + (-0.773 + 0.634i)T \)
11 \( 1 + (0.290 + 0.956i)T \)
13 \( 1 + (-0.634 + 0.773i)T \)
17 \( 1 + (-0.923 + 0.382i)T \)
19 \( 1 + (-0.995 + 0.0980i)T \)
23 \( 1 + (0.980 - 0.195i)T \)
29 \( 1 + (-0.956 - 0.290i)T \)
31 \( 1 + (0.707 + 0.707i)T \)
37 \( 1 + (-0.0980 + 0.995i)T \)
41 \( 1 + (-0.195 - 0.980i)T \)
43 \( 1 + (0.471 - 0.881i)T \)
47 \( 1 + (0.382 + 0.923i)T \)
53 \( 1 + (-0.956 + 0.290i)T \)
59 \( 1 + (0.634 + 0.773i)T \)
61 \( 1 + (-0.881 + 0.471i)T \)
67 \( 1 + (0.881 - 0.471i)T \)
71 \( 1 + (-0.555 - 0.831i)T \)
73 \( 1 + (-0.831 - 0.555i)T \)
79 \( 1 + (0.382 - 0.923i)T \)
83 \( 1 + (-0.0980 - 0.995i)T \)
89 \( 1 + (0.980 + 0.195i)T \)
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

−19.60936800469785305493387647000, −19.16410362530363847691462184607, −18.44469540886793473941749746452, −17.40763000247112914051910297045, −16.97009331186979621075602736954, −15.98403436502248741630516057543, −15.15838453797165327461633938913, −14.622685264189314941246070805999, −13.582762990957975685245788088093, −12.9195870760064853108611705343, −12.52525289680048770357206384325, −11.365021019323741818519345654706, −11.14594333703999378013643123795, −9.647932549553135276801289784684, −8.84793320549383743695702768967, −8.33301208651486857625216054268, −7.57665049377661800823879899556, −6.838655204892322717897310050927, −5.927570575206773538699861918634, −4.976970036062298067860135366608, −3.983282426113681064298416533107, −3.1298816317863754265017672158, −2.27721209506466541307964619245, −1.083365014419318942492201012565, −0.21102717789328524066637232676, 1.89788400484601487169957729943, 2.623596420959398962419370122758, 3.61536240072765553487523280615, 4.41744877270333901484162989947, 4.7804211503237203259626098004, 6.244301150946418965783088770705, 7.03706946432028261304806708104, 7.737689027114376795020605424042, 8.77645437372966519406225392793, 9.24300809223014910290876406562, 10.339051974663279377769772785682, 10.740304180074267967712618841234, 11.64837200596199371361058393019, 12.35796169277897081237290085559, 13.39594833694322761435389482652, 14.31249002185003466001526656406, 14.97488668127453297247293460187, 15.27720391026043321220788244911, 16.09828385562661038890323548561, 17.070571817075855038572479022161, 17.4889851783093260247780122310, 18.92977081055689451988036280948, 19.14953643418061556315453698249, 19.983786210487990691944656620455, 20.6283024330592406588800426747

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