Properties

Label 2-57-57.26-c0-0-0
Degree $2$
Conductor $57$
Sign $0.977 + 0.211i$
Analytic cond. $0.0284467$
Root an. cond. $0.168661$
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  + (−0.5 − 0.866i)3-s + (−0.5 + 0.866i)4-s − 7-s + (−0.499 + 0.866i)9-s + 0.999·12-s + (0.5 − 0.866i)13-s + (−0.499 − 0.866i)16-s + 19-s + (0.5 + 0.866i)21-s + (−0.5 + 0.866i)25-s + 0.999·27-s + (0.5 − 0.866i)28-s − 31-s + (−0.499 − 0.866i)36-s − 37-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)3-s + (−0.5 + 0.866i)4-s − 7-s + (−0.499 + 0.866i)9-s + 0.999·12-s + (0.5 − 0.866i)13-s + (−0.499 − 0.866i)16-s + 19-s + (0.5 + 0.866i)21-s + (−0.5 + 0.866i)25-s + 0.999·27-s + (0.5 − 0.866i)28-s − 31-s + (−0.499 − 0.866i)36-s − 37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 + 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 + 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(57\)    =    \(3 \cdot 19\)
Sign: $0.977 + 0.211i$
Analytic conductor: \(0.0284467\)
Root analytic conductor: \(0.168661\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{57} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 57,\ (\ :0),\ 0.977 + 0.211i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3735359401\)
\(L(\frac12)\) \(\approx\) \(0.3735359401\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 - T \)
good2 \( 1 + (0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T^{2} \)
7 \( 1 + T + T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + T + T^{2} \)
37 \( 1 + T + T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.78158863451418310600567398366, −13.91118790105346893717398966072, −13.04786003793567616204031954427, −12.40570419368847873598634726769, −11.16295984834053954222603330510, −9.530714621167639954748207724266, −8.113025414063899120618495471630, −7.02046440325267567096761130699, −5.50846861495591810506841875830, −3.30843370639598070011740501216, 3.89339273183383705579882973780, 5.42600079494021974315876790764, 6.56213261943062567216093815027, 8.979906840158385872094533206307, 9.755934096435888961389015983965, 10.73867704030288327198980522926, 12.01208341174308721549737477039, 13.52036331503366893986103235457, 14.50422369454586197462514135495, 15.73698933792598545940443005947

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