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Properties

Label 1-2793-2793.1376-r0-0-0
Degree $1$
Conductor $2793$
Sign $0.326 - 0.945i$
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  + (0.623 − 0.781i)2-s + (−0.222 − 0.974i)4-s + (0.900 + 0.433i)5-s + (−0.900 − 0.433i)8-s + (0.900 − 0.433i)10-s + (0.988 + 0.149i)11-s + (−0.365 − 0.930i)13-s + (−0.900 + 0.433i)16-s + (0.733 + 0.680i)17-s + (0.222 − 0.974i)20-s + (0.733 − 0.680i)22-s + (0.733 − 0.680i)23-s + (0.623 + 0.781i)25-s + (−0.955 − 0.294i)26-s + (−0.733 − 0.680i)29-s + ⋯
L(s)  = 1  + (0.623 − 0.781i)2-s + (−0.222 − 0.974i)4-s + (0.900 + 0.433i)5-s + (−0.900 − 0.433i)8-s + (0.900 − 0.433i)10-s + (0.988 + 0.149i)11-s + (−0.365 − 0.930i)13-s + (−0.900 + 0.433i)16-s + (0.733 + 0.680i)17-s + (0.222 − 0.974i)20-s + (0.733 − 0.680i)22-s + (0.733 − 0.680i)23-s + (0.623 + 0.781i)25-s + (−0.955 − 0.294i)26-s + (−0.733 − 0.680i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.326 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.326 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2793\)    =    \(3 \cdot 7^{2} \cdot 19\)
Sign: $0.326 - 0.945i$
Analytic conductor: \(12.9706\)
Root analytic conductor: \(12.9706\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2793} (1376, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2793,\ (0:\ ),\ 0.326 - 0.945i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.498123575 - 1.779323198i\)
\(L(\frac12)\) \(\approx\) \(2.498123575 - 1.779323198i\)
\(L(1)\) \(\approx\) \(1.609292363 - 0.7625010912i\)
\(L(1)\) \(\approx\) \(1.609292363 - 0.7625010912i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.623 - 0.781i)T \)
5 \( 1 + (0.900 + 0.433i)T \)
11 \( 1 + (0.988 + 0.149i)T \)
13 \( 1 + (-0.365 - 0.930i)T \)
17 \( 1 + (0.733 + 0.680i)T \)
23 \( 1 + (0.733 - 0.680i)T \)
29 \( 1 + (-0.733 - 0.680i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (0.733 + 0.680i)T \)
41 \( 1 + (0.0747 + 0.997i)T \)
43 \( 1 + (0.826 + 0.563i)T \)
47 \( 1 + (0.988 + 0.149i)T \)
53 \( 1 + (-0.222 - 0.974i)T \)
59 \( 1 + (0.0747 - 0.997i)T \)
61 \( 1 + (0.955 - 0.294i)T \)
67 \( 1 - T \)
71 \( 1 + (-0.733 + 0.680i)T \)
73 \( 1 + (-0.988 + 0.149i)T \)
79 \( 1 - T \)
83 \( 1 + (-0.623 - 0.781i)T \)
89 \( 1 + (0.365 - 0.930i)T \)
97 \( 1 + (0.5 + 0.866i)T \)
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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

−19.247392896050778947244599815232, −18.54489404150253041778992003263, −17.631392899505023858018459564459, −17.04288311039053987994062046563, −16.63054533520752188400050766902, −15.96145133395643468220244209868, −14.92596237905908266591005368262, −14.36779258926640716317707580633, −13.78821397028589110763056976270, −13.22354272889937035226967600629, −12.27752643690385875508630414552, −11.852880543458264764482006138636, −10.90777500997062250915339405680, −9.637389020392057715653583446942, −9.19731543008163946920798024895, −8.64617383401568341859519606955, −7.32852910335378611914806726331, −7.0881156557816297042471044263, −5.875929566084849752120846613911, −5.67504279024934024118299291901, −4.60421389844155727609761181197, −4.02951874669967629729611378221, −2.982067661968992174064394423412, −2.08908720834858128617212116973, −1.00020833587783520949925397836, 0.95822139585253080851457387069, 1.66627070206080217306095897186, 2.6773664979679196339036052755, 3.20513383452719964698941761074, 4.19378766433262612890827155353, 5.016212303223876446110274718462, 5.86503836738130050998763790355, 6.33474642769757684753648802899, 7.23777929333949186252150061890, 8.39910208545301750113115019028, 9.306648467495962030343017386918, 9.939076824113328678734730158646, 10.44702372294225170363628658674, 11.24637990466229301356809482470, 11.97739744365129531427502232956, 12.890719670056732291920196096577, 13.1450341083973775356347186141, 14.27675261746566433186976171825, 14.59323961198725847397299525666, 15.14138638285907297471304287919, 16.245340729204150646981112346089, 17.35750828966269785580824733091, 17.521778617025489421082146789773, 18.68448703588520329110801941706, 19.02634746750532615853899954872

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