Properties

Label 1-57-57.11-r1-0-0
Degree $1$
Conductor $57$
Sign $-0.671 + 0.740i$
Analytic cond. $6.12550$
Root an. cond. $6.12550$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)4-s + (0.5 + 0.866i)5-s + 7-s − 8-s + (−0.5 + 0.866i)10-s − 11-s + (−0.5 + 0.866i)13-s + (0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)17-s − 20-s + (−0.5 − 0.866i)22-s + (0.5 − 0.866i)23-s + (−0.5 + 0.866i)25-s − 26-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)4-s + (0.5 + 0.866i)5-s + 7-s − 8-s + (−0.5 + 0.866i)10-s − 11-s + (−0.5 + 0.866i)13-s + (0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)17-s − 20-s + (−0.5 − 0.866i)22-s + (0.5 − 0.866i)23-s + (−0.5 + 0.866i)25-s − 26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.671 + 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.671 + 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(57\)    =    \(3 \cdot 19\)
Sign: $-0.671 + 0.740i$
Analytic conductor: \(6.12550\)
Root analytic conductor: \(6.12550\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{57} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 57,\ (1:\ ),\ -0.671 + 0.740i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8119985845 + 1.832188669i\)
\(L(\frac12)\) \(\approx\) \(0.8119985845 + 1.832188669i\)
\(L(1)\) \(\approx\) \(1.067915846 + 0.9681772550i\)
\(L(1)\) \(\approx\) \(1.067915846 + 0.9681772550i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + T \)
11 \( 1 - T \)
13 \( 1 + (-0.5 + 0.866i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 + T \)
37 \( 1 + T \)
41 \( 1 + (0.5 + 0.866i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (0.5 - 0.866i)T \)
59 \( 1 + (0.5 + 0.866i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 - T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (-0.5 - 0.866i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−31.97875476826675965947481656666, −31.22395769943453396678035535278, −29.89275706866518263573340082269, −29.03947359003603324621965388867, −27.92347278730336446017000157651, −27.082588301456514617246007932331, −25.1460887972799216650389048666, −24.12520899658294230668393126305, −23.11330234908604359073834969944, −21.573760275458204909418201753449, −20.859563669931778789989846291952, −19.972762737774123698712663276761, −18.35498629680616734541451677802, −17.41383123532230284238353038478, −15.610684052661416659955322483643, −14.22140615370001944579598641439, −13.135214330097599173786759724, −12.06575007244866482812135002651, −10.69710220749105265238668222275, −9.47245055206400385515295207352, −7.982326679377617106665930952676, −5.48545729013457190449960040377, −4.77321373331126485023358308620, −2.67098374995128088560317158195, −1.02702521794865644540331165104, 2.5488432379526665452185424833, 4.4759917534321526622685442712, 5.84944553066873169228082911939, 7.20136394785676526263100763442, 8.39054945198850491026151301607, 10.199401347217629675152279606701, 11.72991160709419412807743232294, 13.31584862648504745166393665994, 14.42577606456676618430191017742, 15.168212602889338865820207336793, 16.77751512088941891006265840560, 17.81850021202068781592838928715, 18.811500162662461169481304242461, 21.05634230838466790642720891109, 21.628779321650693107866161515676, 23.01326464575640717671396774324, 23.97260210265733261574785231630, 25.08166683644268962194945038140, 26.30543871366964565222571736461, 26.90137077488091019530496316096, 28.62560565435512977619231895651, 30.16154848132711754910970876969, 30.841569436562295669636982343183, 32.00303150340570409662392255355, 33.37459266749032514818082899074

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