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Properties

Label 1-2793-2793.1409-r0-0-0
Degree $1$
Conductor $2793$
Sign $0.894 + 0.447i$
Analytic cond. $12.9706$
Root an. cond. $12.9706$
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  + (0.980 + 0.198i)2-s + (0.921 + 0.388i)4-s + (0.583 + 0.811i)5-s + (0.826 + 0.563i)8-s + (0.411 + 0.911i)10-s + (−0.623 − 0.781i)11-s + (0.0249 − 0.999i)13-s + (0.698 + 0.715i)16-s + (0.124 − 0.992i)17-s + (0.222 + 0.974i)20-s + (−0.456 − 0.889i)22-s + (0.797 − 0.603i)23-s + (−0.318 + 0.947i)25-s + (0.222 − 0.974i)26-s + (−0.998 + 0.0498i)29-s + ⋯
L(s)  = 1  + (0.980 + 0.198i)2-s + (0.921 + 0.388i)4-s + (0.583 + 0.811i)5-s + (0.826 + 0.563i)8-s + (0.411 + 0.911i)10-s + (−0.623 − 0.781i)11-s + (0.0249 − 0.999i)13-s + (0.698 + 0.715i)16-s + (0.124 − 0.992i)17-s + (0.222 + 0.974i)20-s + (−0.456 − 0.889i)22-s + (0.797 − 0.603i)23-s + (−0.318 + 0.947i)25-s + (0.222 − 0.974i)26-s + (−0.998 + 0.0498i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2793\)    =    \(3 \cdot 7^{2} \cdot 19\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(12.9706\)
Root analytic conductor: \(12.9706\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2793} (1409, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2793,\ (0:\ ),\ 0.894 + 0.447i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.784662203 + 0.8935406993i\)
\(L(\frac12)\) \(\approx\) \(3.784662203 + 0.8935406993i\)
\(L(1)\) \(\approx\) \(2.158868952 + 0.4008227975i\)
\(L(1)\) \(\approx\) \(2.158868952 + 0.4008227975i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.980 + 0.198i)T \)
5 \( 1 + (0.583 + 0.811i)T \)
11 \( 1 + (-0.623 - 0.781i)T \)
13 \( 1 + (0.0249 - 0.999i)T \)
17 \( 1 + (0.124 - 0.992i)T \)
23 \( 1 + (0.797 - 0.603i)T \)
29 \( 1 + (-0.998 + 0.0498i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (0.733 - 0.680i)T \)
41 \( 1 + (-0.411 + 0.911i)T \)
43 \( 1 + (0.698 + 0.715i)T \)
47 \( 1 + (0.853 + 0.521i)T \)
53 \( 1 + (0.921 + 0.388i)T \)
59 \( 1 + (0.698 + 0.715i)T \)
61 \( 1 + (0.456 - 0.889i)T \)
67 \( 1 + (-0.766 - 0.642i)T \)
71 \( 1 + (0.542 - 0.840i)T \)
73 \( 1 + (0.878 + 0.478i)T \)
79 \( 1 + (0.939 + 0.342i)T \)
83 \( 1 + (-0.365 - 0.930i)T \)
89 \( 1 + (-0.661 - 0.749i)T \)
97 \( 1 + (-0.173 - 0.984i)T \)
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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

−19.27190235941229211928787321116, −18.67796662415720670743805025087, −17.5587974743141693503260822576, −16.89847497779788064049729092343, −16.393654683724317880291511756595, −15.35505538477847121041623176235, −15.02660161651133223057982244308, −14.02565063430833458086447154946, −13.38312144896350501002732213988, −12.90630995711144554874025102000, −12.20107879234470162550606322513, −11.52980121440351057590255138282, −10.634876132416536600543601058130, −9.90602970848861137538720783782, −9.24955938806863709246989922507, −8.26255678942122698418227922834, −7.33209588779239404699556748260, −6.62046868499142510328882331928, −5.656769421141746056084459386583, −5.24748638682387616268502592203, −4.28862927483708571476919934238, −3.81762006881446449175010690295, −2.43832174054993944814924656318, −1.976013235540507242767267077573, −1.05289589189023311218693296393, 0.97759322484489408510179169460, 2.32406651839747386641969409201, 2.89828302870251976241497805888, 3.420171016629541589803943637081, 4.61198412298529792867929135931, 5.44419794341392264726282548444, 5.8893929950640886362280919763, 6.76381901254827214792858531676, 7.45685929223134789031551084140, 8.1433501180370925183819269326, 9.22671245235877725153445487367, 10.21438965081257788927410158409, 10.88701497147790316092599831362, 11.288125400562323281724448340775, 12.360516425347251398399933997880, 13.08120092653554035474389183072, 13.610023291966038254934587595733, 14.291416208893483902436684995334, 14.937551193120355399146846175749, 15.58147344547891283880800376077, 16.331424582365381068391620424039, 17.00331331102734224215422636488, 17.9379207768167789835452060840, 18.45320700758079505551237715479, 19.32169925017922958096799477501

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