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Properties

Label 1-1792-1792.1171-r1-0-0
Degree $1$
Conductor $1792$
Sign $-0.809 - 0.587i$
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  + (0.849 − 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.809 - 0.587i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.809 - 0.587i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $-0.809 - 0.587i$
Analytic conductor: \(192.577\)
Root analytic conductor: \(192.577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (1171, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (1:\ ),\ -0.809 - 0.587i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5316289568 - 1.637406861i\)
\(L(\frac12)\) \(\approx\) \(0.5316289568 - 1.637406861i\)
\(L(1)\) \(\approx\) \(1.093316408 - 0.3843458245i\)
\(L(1)\) \(\approx\) \(1.093316408 - 0.3843458245i\)

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

−20.18209408447661098352328759815, −19.83763778576592768066673770256, −19.0216067063623205433624321284, −18.25246949067602959872806077163, −17.357599624674845408030888409266, −16.336326201123713093057332833042, −15.84465118587533258877642715984, −15.08577124350576383936295228388, −14.489731342143747829583622465347, −13.958956148025518721695613442438, −12.6325646690089521903348171127, −12.33140297120891717809803196357, −11.16151646915928894551492715030, −10.584312340124990285955618687287, −9.602044715803276159973717250596, −9.01162577026526310885371635687, −8.19447140878036475965122328621, −7.28116491237592192349432443346, −6.99301038138831555426749210257, −5.394209362867450824395783336755, −4.57993128481370378452707509856, −3.95322840147135339756101107813, −3.05282604187039329013357485267, −2.353943114335226437504730458713, −1.03609781351017760574039399500, 0.30591999684342399881273365105, 1.25584164226945211063475369605, 2.292745359134358488191476615327, 3.27635472779817788051076663417, 3.95268914347556383770384913410, 4.725938736371999147163323319339, 6.068852973232175982455448134956, 6.87172180629786123725377210428, 7.637951976215759115681382472077, 8.16743414057587673753409577965, 9.24502243213983641610190083215, 9.41910931194097873344749861503, 10.920818957634078654818976340669, 11.70482753796890810173121784585, 12.10843155678933353152002850047, 13.23582581656549541766917104603, 13.6473325061216713143011560441, 14.66819299709495586624571433848, 15.14839864752013599743511956714, 15.88645154772803235611815074192, 16.82470306841092342759686282256, 17.49754599789996768499759217476, 18.578496599844679214640518874289, 19.14436370555801677712806454260, 19.68084296368685763354481195431

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