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Properties

Label 1-793-793.302-r0-0-0
Degree $1$
Conductor $793$
Sign $0.869 - 0.494i$
Analytic cond. $3.68267$
Root an. cond. $3.68267$
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  + (0.913 − 0.406i)2-s + (0.913 − 0.406i)3-s + (0.669 − 0.743i)4-s + (0.309 + 0.951i)5-s + (0.669 − 0.743i)6-s + (0.913 + 0.406i)7-s + (0.309 − 0.951i)8-s + (0.669 − 0.743i)9-s + (0.669 + 0.743i)10-s + (−0.5 + 0.866i)11-s + (0.309 − 0.951i)12-s + 14-s + (0.669 + 0.743i)15-s + (−0.104 − 0.994i)16-s + (0.669 − 0.743i)17-s + (0.309 − 0.951i)18-s + ⋯
L(s)  = 1  + (0.913 − 0.406i)2-s + (0.913 − 0.406i)3-s + (0.669 − 0.743i)4-s + (0.309 + 0.951i)5-s + (0.669 − 0.743i)6-s + (0.913 + 0.406i)7-s + (0.309 − 0.951i)8-s + (0.669 − 0.743i)9-s + (0.669 + 0.743i)10-s + (−0.5 + 0.866i)11-s + (0.309 − 0.951i)12-s + 14-s + (0.669 + 0.743i)15-s + (−0.104 − 0.994i)16-s + (0.669 − 0.743i)17-s + (0.309 − 0.951i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 793 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.869 - 0.494i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 793 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.869 - 0.494i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(793\)    =    \(13 \cdot 61\)
Sign: $0.869 - 0.494i$
Analytic conductor: \(3.68267\)
Root analytic conductor: \(3.68267\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{793} (302, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 793,\ (0:\ ),\ 0.869 - 0.494i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.879311751 - 1.026782887i\)
\(L(\frac12)\) \(\approx\) \(3.879311751 - 1.026782887i\)
\(L(1)\) \(\approx\) \(2.556720073 - 0.5642355845i\)
\(L(1)\) \(\approx\) \(2.556720073 - 0.5642355845i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
61 \( 1 \)
good2 \( 1 + (0.913 - 0.406i)T \)
3 \( 1 + (0.913 - 0.406i)T \)
5 \( 1 + (0.309 + 0.951i)T \)
7 \( 1 + (0.913 + 0.406i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
17 \( 1 + (0.669 - 0.743i)T \)
19 \( 1 + (-0.104 + 0.994i)T \)
23 \( 1 + (-0.978 - 0.207i)T \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + (-0.809 + 0.587i)T \)
37 \( 1 + (-0.104 - 0.994i)T \)
41 \( 1 + (-0.104 - 0.994i)T \)
43 \( 1 + (-0.978 + 0.207i)T \)
47 \( 1 + T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (-0.104 + 0.994i)T \)
67 \( 1 + (-0.978 - 0.207i)T \)
71 \( 1 + (-0.978 + 0.207i)T \)
73 \( 1 + (0.309 - 0.951i)T \)
79 \( 1 + (0.309 - 0.951i)T \)
83 \( 1 + (-0.809 - 0.587i)T \)
89 \( 1 + (0.913 - 0.406i)T \)
97 \( 1 + (0.913 + 0.406i)T \)
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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

−21.87589818984625536037612735811, −21.66802249998748587010705601508, −20.77132853676184302211828338309, −20.36029666527341402650007367378, −19.53232733735687535773548816113, −18.29239432411479236368553868707, −17.071613031967562655171572333494, −16.654563202779032771022489023468, −15.66654783745147089567555282188, −15.03623807602344505933369505824, −14.09345265000894574284337449161, −13.523151280850936545695803299899, −12.97150737439720393240770020976, −11.81148275356393604001022268282, −10.91691622054041516485971161544, −9.9064339345483264358087042243, −8.67630861522538558113226617483, −8.13334328109838628291229317041, −7.48114369405651295754408519866, −5.99656278244498463102672663200, −5.17923093552275593562727650442, −4.389779242879933154676051092427, −3.64478476397052440196678863739, −2.45737792150870688861992078969, −1.495432097029216426372256008727, 1.69984358242754306313635553173, 2.12246914343429877209584800884, 3.11631465510419438079115521832, 3.96806944154477496113022827580, 5.1618910348348020262881479705, 6.02349094399416755634369001702, 7.26293980967776275163852476523, 7.58811027043159040540443407414, 8.98600779438694618717709819207, 10.05969994876770383833302947296, 10.61143014311255518376037235057, 11.8562456778749002736912115606, 12.37222468467602483025512261674, 13.395735510765598280706933136603, 14.31288869768137092738399670322, 14.544331653307210498532572627667, 15.26265163458068505996777286112, 16.22959329953039013956141510152, 17.91267047270200519886198701182, 18.335333734348687136004627293135, 19.025230696491611632774570402741, 20.08903515454760102198921079116, 20.71094973388286864800179499994, 21.30825068674202035264618704696, 22.12692689121075156842393988557

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