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Properties

Label 1-1792-1792.1157-r0-0-0
Degree $1$
Conductor $1792$
Sign $-0.959 - 0.280i$
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  + (−0.729 + 0.683i)3-s + (0.412 + 0.910i)5-s + (0.0654 − 0.997i)9-s + (0.528 + 0.849i)11-s + (0.0980 + 0.995i)13-s + (−0.923 − 0.382i)15-s + (0.130 − 0.991i)17-s + (−0.986 − 0.162i)19-s + (0.751 − 0.659i)23-s + (−0.659 + 0.751i)25-s + (0.634 + 0.773i)27-s + (−0.881 + 0.471i)29-s + (−0.965 + 0.258i)31-s + (−0.965 − 0.258i)33-s + (−0.935 − 0.352i)37-s + ⋯
L(s)  = 1  + (−0.729 + 0.683i)3-s + (0.412 + 0.910i)5-s + (0.0654 − 0.997i)9-s + (0.528 + 0.849i)11-s + (0.0980 + 0.995i)13-s + (−0.923 − 0.382i)15-s + (0.130 − 0.991i)17-s + (−0.986 − 0.162i)19-s + (0.751 − 0.659i)23-s + (−0.659 + 0.751i)25-s + (0.634 + 0.773i)27-s + (−0.881 + 0.471i)29-s + (−0.965 + 0.258i)31-s + (−0.965 − 0.258i)33-s + (−0.935 − 0.352i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.959 - 0.280i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.959 - 0.280i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $-0.959 - 0.280i$
Analytic conductor: \(8.32201\)
Root analytic conductor: \(8.32201\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (1157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (0:\ ),\ -0.959 - 0.280i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.09231606676 + 0.6451195853i\)
\(L(\frac12)\) \(\approx\) \(-0.09231606676 + 0.6451195853i\)
\(L(1)\) \(\approx\) \(0.6843503312 + 0.3911856096i\)
\(L(1)\) \(\approx\) \(0.6843503312 + 0.3911856096i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.729 + 0.683i)T \)
5 \( 1 + (0.412 + 0.910i)T \)
11 \( 1 + (0.528 + 0.849i)T \)
13 \( 1 + (0.0980 + 0.995i)T \)
17 \( 1 + (0.130 - 0.991i)T \)
19 \( 1 + (-0.986 - 0.162i)T \)
23 \( 1 + (0.751 - 0.659i)T \)
29 \( 1 + (-0.881 + 0.471i)T \)
31 \( 1 + (-0.965 + 0.258i)T \)
37 \( 1 + (-0.935 - 0.352i)T \)
41 \( 1 + (0.980 - 0.195i)T \)
43 \( 1 + (-0.956 + 0.290i)T \)
47 \( 1 + (-0.608 + 0.793i)T \)
53 \( 1 + (0.849 - 0.528i)T \)
59 \( 1 + (0.812 + 0.582i)T \)
61 \( 1 + (-0.973 + 0.227i)T \)
67 \( 1 + (-0.683 - 0.729i)T \)
71 \( 1 + (-0.831 + 0.555i)T \)
73 \( 1 + (0.442 + 0.896i)T \)
79 \( 1 + (-0.991 + 0.130i)T \)
83 \( 1 + (0.773 + 0.634i)T \)
89 \( 1 + (-0.946 + 0.321i)T \)
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

−19.59180741214077285248887128091, −19.16690781218386393782390959499, −18.22698493428911456742365502924, −17.45555970904033753540905537109, −16.89857493956401919778349055081, −16.51281387320209272437690561148, −15.433099206465157614313828049102, −14.603528596988388013679840655823, −13.41763602195294869867654264064, −13.15845187956976786531603810769, −12.44605000461912979651497859296, −11.65208940996789298818367123218, −10.8443925235243959235802241651, −10.17855523206501821895888349047, −9.03110420029216304767149491530, −8.38429657349231201099172581823, −7.65173328889327040771603742999, −6.58195462083783516749832014433, −5.7662425533250272632405813543, −5.450103646109844954938834563663, −4.33214140513018238836367884577, −3.3453059879844418547467884447, −1.96977339863079525519720939638, −1.32711970526829532993950599747, −0.25596927275355428641593979874, 1.479587746911630252929930219395, 2.459202722350088291356530784412, 3.53671730414676829693183725722, 4.33085092207072864775506070037, 5.1207352153780702830989882755, 6.0466943004647223296727223846, 6.91024417403545584998818023874, 7.15785120143265565375046583047, 8.86082383709282248195197406897, 9.37850797410029626466804203863, 10.12892264658935538964919080645, 10.922867863982199228113869361658, 11.41367585626232500763015679279, 12.26704950552527273291159999539, 13.09727082396415561190549737219, 14.267562705922217027548717245290, 14.680746889775541466844161835722, 15.34891178105186343233142692987, 16.37490198748378178193420001282, 16.84549538392543676193645005346, 17.70153916274984838423527695478, 18.24376857192655176549408188355, 18.99321543139539031658527316442, 19.90132600566078347916292698141, 20.953838455642914946094800703785

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