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Properties

Label 1-2793-2793.452-r0-0-0
Degree $1$
Conductor $2793$
Sign $-0.963 - 0.268i$
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  + (0.698 − 0.715i)2-s + (−0.0249 − 0.999i)4-s + (0.797 − 0.603i)5-s + (−0.733 − 0.680i)8-s + (0.124 − 0.992i)10-s + (0.900 − 0.433i)11-s + (−0.995 + 0.0995i)13-s + (−0.998 + 0.0498i)16-s + (−0.878 + 0.478i)17-s + (−0.623 − 0.781i)20-s + (0.318 − 0.947i)22-s + (0.853 − 0.521i)23-s + (0.270 − 0.962i)25-s + (−0.623 + 0.781i)26-s + (0.980 + 0.198i)29-s + ⋯
L(s)  = 1  + (0.698 − 0.715i)2-s + (−0.0249 − 0.999i)4-s + (0.797 − 0.603i)5-s + (−0.733 − 0.680i)8-s + (0.124 − 0.992i)10-s + (0.900 − 0.433i)11-s + (−0.995 + 0.0995i)13-s + (−0.998 + 0.0498i)16-s + (−0.878 + 0.478i)17-s + (−0.623 − 0.781i)20-s + (0.318 − 0.947i)22-s + (0.853 − 0.521i)23-s + (0.270 − 0.962i)25-s + (−0.623 + 0.781i)26-s + (0.980 + 0.198i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.963 - 0.268i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.963 - 0.268i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2793\)    =    \(3 \cdot 7^{2} \cdot 19\)
Sign: $-0.963 - 0.268i$
Analytic conductor: \(12.9706\)
Root analytic conductor: \(12.9706\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2793} (452, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2793,\ (0:\ ),\ -0.963 - 0.268i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3423685596 - 2.507460673i\)
\(L(\frac12)\) \(\approx\) \(0.3423685596 - 2.507460673i\)
\(L(1)\) \(\approx\) \(1.215173907 - 1.094650517i\)
\(L(1)\) \(\approx\) \(1.215173907 - 1.094650517i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.698 - 0.715i)T \)
5 \( 1 + (0.797 - 0.603i)T \)
11 \( 1 + (0.900 - 0.433i)T \)
13 \( 1 + (-0.995 + 0.0995i)T \)
17 \( 1 + (-0.878 + 0.478i)T \)
23 \( 1 + (0.853 - 0.521i)T \)
29 \( 1 + (0.980 + 0.198i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (0.988 - 0.149i)T \)
41 \( 1 + (-0.124 - 0.992i)T \)
43 \( 1 + (-0.998 + 0.0498i)T \)
47 \( 1 + (0.583 + 0.811i)T \)
53 \( 1 + (-0.0249 - 0.999i)T \)
59 \( 1 + (-0.998 + 0.0498i)T \)
61 \( 1 + (-0.318 - 0.947i)T \)
67 \( 1 + (0.939 + 0.342i)T \)
71 \( 1 + (-0.661 - 0.749i)T \)
73 \( 1 + (-0.411 - 0.911i)T \)
79 \( 1 + (-0.173 + 0.984i)T \)
83 \( 1 + (-0.0747 - 0.997i)T \)
89 \( 1 + (-0.969 + 0.246i)T \)
97 \( 1 + (-0.766 - 0.642i)T \)
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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

−19.735827393654214572419565211, −18.61050984870820045641342701451, −17.87859100813179268408081896721, −17.27985926286564099733643985761, −16.89672100458898339143453975931, −15.88979064847963104946308990716, −15.02608472968311629440611797477, −14.74858317735971788472979820090, −13.853234377706893514101876561454, −13.45532584660062766459776150741, −12.55050261495307521038073822016, −11.84476352443279635747990437955, −11.153496422324754153904371987790, −10.099458827952214433270744122735, −9.39852877489081373881132498702, −8.71823244035882049415396885465, −7.65913798240669212084547990132, −6.80427477468414847672853528617, −6.6385708535091830802985881687, −5.60040935511478559692554394771, −4.842309309475478059759013234048, −4.20380157573138844712953659220, −2.98492317898736441975778003605, −2.58610282523337214339472314747, −1.42213607539465770380445362680, 0.58255376342132951719543504240, 1.527710755414235855612016201184, 2.30654459060206674734208098728, 3.057049789185571167016172445853, 4.24239255985213798262716513994, 4.666658976883466754778209950003, 5.52872305694695974910401648114, 6.3385156053187099230313291521, 6.83450584901840886699138668288, 8.25121053536570841778621810917, 9.09520654766890430277189426548, 9.549440982575412863967935540360, 10.371363268775478093265403459372, 11.09424178900588987477487147180, 11.920715921514961606786474602, 12.521953658202837870737824015136, 13.18513073511158327593990651529, 13.83820493489101401130417029773, 14.48348197505905280781282280705, 15.11278208360697537211069022586, 16.0326344355768360707546154465, 16.963770999427403660028049833, 17.37812225314010070491998602024, 18.30682662352397637414954056454, 19.15585540002637464131699708274

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