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Properties

Label 1-57-57.41-r0-0-0
Degree $1$
Conductor $57$
Sign $-0.174 + 0.984i$
Analytic cond. $0.264706$
Root an. cond. $0.264706$
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  + (0.173 + 0.984i)2-s + (−0.939 + 0.342i)4-s + (0.939 + 0.342i)5-s + (−0.5 + 0.866i)7-s + (−0.5 − 0.866i)8-s + (−0.173 + 0.984i)10-s + (0.5 + 0.866i)11-s + (−0.766 − 0.642i)13-s + (−0.939 − 0.342i)14-s + (0.766 − 0.642i)16-s + (−0.173 − 0.984i)17-s − 20-s + (−0.766 + 0.642i)22-s + (0.939 − 0.342i)23-s + (0.766 + 0.642i)25-s + (0.5 − 0.866i)26-s + ⋯
L(s)  = 1  + (0.173 + 0.984i)2-s + (−0.939 + 0.342i)4-s + (0.939 + 0.342i)5-s + (−0.5 + 0.866i)7-s + (−0.5 − 0.866i)8-s + (−0.173 + 0.984i)10-s + (0.5 + 0.866i)11-s + (−0.766 − 0.642i)13-s + (−0.939 − 0.342i)14-s + (0.766 − 0.642i)16-s + (−0.173 − 0.984i)17-s − 20-s + (−0.766 + 0.642i)22-s + (0.939 − 0.342i)23-s + (0.766 + 0.642i)25-s + (0.5 − 0.866i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.174 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.174 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(57\)    =    \(3 \cdot 19\)
Sign: $-0.174 + 0.984i$
Analytic conductor: \(0.264706\)
Root analytic conductor: \(0.264706\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{57} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 57,\ (0:\ ),\ -0.174 + 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6077953449 + 0.7248225467i\)
\(L(\frac12)\) \(\approx\) \(0.6077953449 + 0.7248225467i\)
\(L(1)\) \(\approx\) \(0.8667234833 + 0.6187931938i\)
\(L(1)\) \(\approx\) \(0.8667234833 + 0.6187931938i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.173 + 0.984i)T \)
5 \( 1 + (0.939 + 0.342i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (-0.766 - 0.642i)T \)
17 \( 1 + (-0.173 - 0.984i)T \)
23 \( 1 + (0.939 - 0.342i)T \)
29 \( 1 + (0.173 - 0.984i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 - T \)
41 \( 1 + (0.766 - 0.642i)T \)
43 \( 1 + (-0.939 - 0.342i)T \)
47 \( 1 + (-0.173 + 0.984i)T \)
53 \( 1 + (-0.939 + 0.342i)T \)
59 \( 1 + (0.173 + 0.984i)T \)
61 \( 1 + (-0.939 + 0.342i)T \)
67 \( 1 + (-0.173 + 0.984i)T \)
71 \( 1 + (-0.939 - 0.342i)T \)
73 \( 1 + (0.766 - 0.642i)T \)
79 \( 1 + (-0.766 + 0.642i)T \)
83 \( 1 + (0.5 - 0.866i)T \)
89 \( 1 + (0.766 + 0.642i)T \)
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

−32.57637132692121609218641907323, −31.49617754927088811083138206526, −30.01252662001282626912823223958, −29.37153805030093941494657499301, −28.52061249393585102429112572465, −27.082091569368029063124174743526, −26.13824452907088080542437243541, −24.51866768605796804806670313681, −23.35205339541211711847911886525, −21.96866382478602605973663556863, −21.30230994874703382983453916105, −19.94460725561293330310895586913, −19.10739717443278879797421764028, −17.520822273450884887224292282074, −16.67218854982305020240350808334, −14.42902316119806014162908844782, −13.54134656864630331000437040735, −12.50747860760458802056657309859, −10.93668149189960526621570116012, −9.852038647010820539245450152374, −8.75784792740989738424037575652, −6.483191541039964732611343358036, −4.88174169296693573692084506043, −3.31946694304791485919382994952, −1.475562779783102637766625746055, 2.72984526246235140767908668517, 4.89890724724105088675221851574, 6.114560457652120145987460866172, 7.26358900431775537839898248888, 9.075609591594104480090949829279, 9.914482947113240477285222565011, 12.20675917906107073595582094982, 13.34430131329149329136470919047, 14.63466326013451613180774809777, 15.52232901507429117980688754955, 17.04458715726933389795587854491, 17.89258795532597902830601364634, 19.0490108901227497937498717808, 20.94828960804328665618860159013, 22.347550134087659727931456431889, 22.68367261083105506071938291166, 24.686400262169207605574780363474, 25.12320566813421134975610222927, 26.1656481140313772885974404716, 27.429059141046289400719104958136, 28.69173770478102770660959590617, 30.01819112915587646756581950462, 31.2923123239067786000201799463, 32.32492135635258141239960093950, 33.285343262210117306952016833

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