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Properties

Label 2-57-57.8-c9-0-28
Degree $2$
Conductor $57$
Sign $0.0852 + 0.996i$
Analytic cond. $29.3570$
Root an. cond. $5.41821$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−121.5 − 70.1i)3-s + (256 + 443. i)4-s − 7.82e3·7-s + (9.84e3 + 1.70e4i)9-s − 7.18e4i·12-s + (−1.57e4 + 9.12e3i)13-s + (−1.31e5 + 2.27e5i)16-s + (4.88e5 + 2.90e5i)19-s + (9.51e5 + 5.49e5i)21-s + (−9.76e5 − 1.69e6i)25-s − 2.76e6i·27-s + (−2.00e6 − 3.47e6i)28-s − 6.53e6i·31-s + (−5.03e6 + 8.72e6i)36-s − 2.17e7i·37-s + ⋯
L(s)  = 1  + (−0.866 − 0.500i)3-s + (0.5 + 0.866i)4-s − 1.23·7-s + (0.499 + 0.866i)9-s − 1.00i·12-s + (−0.153 + 0.0885i)13-s + (−0.499 + 0.866i)16-s + (0.859 + 0.510i)19-s + (1.06 + 0.616i)21-s + (−0.5 − 0.866i)25-s − 0.999i·27-s + (−0.616 − 1.06i)28-s − 1.27i·31-s + (−0.499 + 0.866i)36-s − 1.90i·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0852 + 0.996i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.0852 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(57\)    =    \(3 \cdot 19\)
Sign: $0.0852 + 0.996i$
Analytic conductor: \(29.3570\)
Root analytic conductor: \(5.41821\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{57} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 57,\ (\ :9/2),\ 0.0852 + 0.996i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.570671 - 0.523935i\)
\(L(\frac12)\) \(\approx\) \(0.570671 - 0.523935i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (121.5 + 70.1i)T \)
19 \( 1 + (-4.88e5 - 2.90e5i)T \)
good2 \( 1 + (-256 - 443. i)T^{2} \)
5 \( 1 + (9.76e5 + 1.69e6i)T^{2} \)
7 \( 1 + 7.82e3T + 4.03e7T^{2} \)
11 \( 1 - 2.35e9T^{2} \)
13 \( 1 + (1.57e4 - 9.12e3i)T + (5.30e9 - 9.18e9i)T^{2} \)
17 \( 1 + (5.92e10 + 1.02e11i)T^{2} \)
23 \( 1 + (9.00e11 - 1.55e12i)T^{2} \)
29 \( 1 + (-7.25e12 + 1.25e13i)T^{2} \)
31 \( 1 + 6.53e6iT - 2.64e13T^{2} \)
37 \( 1 + 2.17e7iT - 1.29e14T^{2} \)
41 \( 1 + (-1.63e14 - 2.83e14i)T^{2} \)
43 \( 1 + (-1.38e7 + 2.40e7i)T + (-2.51e14 - 4.35e14i)T^{2} \)
47 \( 1 + (5.59e14 - 9.69e14i)T^{2} \)
53 \( 1 + (-1.64e15 + 2.85e15i)T^{2} \)
59 \( 1 + (-4.33e15 - 7.50e15i)T^{2} \)
61 \( 1 + (4.90e7 + 8.49e7i)T + (-5.84e15 + 1.01e16i)T^{2} \)
67 \( 1 + (-2.18e8 + 1.26e8i)T + (1.36e16 - 2.35e16i)T^{2} \)
71 \( 1 + (-2.29e16 - 3.97e16i)T^{2} \)
73 \( 1 + (9.22e7 - 1.59e8i)T + (-2.94e16 - 5.09e16i)T^{2} \)
79 \( 1 + (-5.98e8 - 3.45e8i)T + (5.99e16 + 1.03e17i)T^{2} \)
83 \( 1 - 1.86e17T^{2} \)
89 \( 1 + (-1.75e17 + 3.03e17i)T^{2} \)
97 \( 1 + (1.01e9 + 5.87e8i)T + (3.80e17 + 6.58e17i)T^{2} \)
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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

−12.69343615539276038784406373624, −12.09931225045547378960535901493, −10.94315409830250265999285602584, −9.615111296069645716728044563040, −7.85879796324620778338816844789, −6.85700788209995593929391627478, −5.81619074649874957453237445884, −3.85106554334753250899310433681, −2.29324114891352498085098225679, −0.31426759905264927185373039777, 1.06634177622684301320919154011, 3.16396657446863793702191739907, 4.98882645662554370317699323420, 6.11334904115822749487478019546, 7.02740813043268387651911947812, 9.392803008146507813752950697527, 10.07397269224408230376241003339, 11.16151538963473168653615109243, 12.19096301021078688615588365108, 13.49964686470248674071385462455

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