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Properties

Label 2-57-19.12-c2-0-5
Degree $2$
Conductor $57$
Sign $0.676 + 0.736i$
Analytic cond. $1.55313$
Root an. cond. $1.24624$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.08 − 1.78i)2-s + (−1.5 + 0.866i)3-s + (4.34 − 7.51i)4-s + (2.32 + 4.03i)5-s + (−3.08 + 5.34i)6-s − 10.6·7-s − 16.6i·8-s + (1.5 − 2.59i)9-s + (14.3 + 8.28i)10-s − 6.37·11-s + 15.0i·12-s + (15.3 + 8.87i)13-s + (−32.9 + 19.0i)14-s + (−6.98 − 4.03i)15-s + (−12.3 − 21.3i)16-s + (−5.84 − 10.1i)17-s + ⋯
L(s)  = 1  + (1.54 − 0.890i)2-s + (−0.5 + 0.288i)3-s + (1.08 − 1.87i)4-s + (0.465 + 0.806i)5-s + (−0.514 + 0.890i)6-s − 1.52·7-s − 2.08i·8-s + (0.166 − 0.288i)9-s + (1.43 + 0.828i)10-s − 0.579·11-s + 1.25i·12-s + (1.18 + 0.682i)13-s + (−2.35 + 1.35i)14-s + (−0.465 − 0.268i)15-s + (−0.770 − 1.33i)16-s + (−0.344 − 0.596i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.676 + 0.736i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.676 + 0.736i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(57\)    =    \(3 \cdot 19\)
Sign: $0.676 + 0.736i$
Analytic conductor: \(1.55313\)
Root analytic conductor: \(1.24624\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{57} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 57,\ (\ :1),\ 0.676 + 0.736i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.88436 - 0.827411i\)
\(L(\frac12)\) \(\approx\) \(1.88436 - 0.827411i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.5 - 0.866i)T \)
19 \( 1 + (3.88 - 18.5i)T \)
good2 \( 1 + (-3.08 + 1.78i)T + (2 - 3.46i)T^{2} \)
5 \( 1 + (-2.32 - 4.03i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + 10.6T + 49T^{2} \)
11 \( 1 + 6.37T + 121T^{2} \)
13 \( 1 + (-15.3 - 8.87i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + (5.84 + 10.1i)T + (-144.5 + 250. i)T^{2} \)
23 \( 1 + (-13.9 + 24.2i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (33.2 + 19.1i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + 42.9iT - 961T^{2} \)
37 \( 1 - 33.9iT - 1.36e3T^{2} \)
41 \( 1 + (-16.3 + 9.45i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-26.5 - 45.9i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (12.0 - 20.8i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (13.3 + 7.69i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-25.6 + 14.8i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (21.3 - 36.9i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (15.1 + 8.75i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (74.6 - 43.0i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (-46.2 - 80.0i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (26.3 - 15.2i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 77.1T + 6.88e3T^{2} \)
89 \( 1 + (76.8 + 44.3i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (1.82 - 1.05i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−14.53928234665616236216787334556, −13.42067324922634166326163424132, −12.80051542379794047563604541906, −11.46516795833986393253823767525, −10.58391062378850390853701182165, −9.652094205226805965621244906145, −6.53592337228709977535872297241, −5.91927260795359391394277318020, −4.07565409463688850198707509694, −2.74861974024507576704049969062, 3.43232400942395974462191543339, 5.23840760169883637494100121473, 6.07238600745051223249400648572, 7.19969846702714987664600363578, 8.970489383779305477338852812667, 10.86634393555356655272126139884, 12.55905567433166621903327075068, 13.05110238003502674542080385425, 13.54700403179236919458647866949, 15.33702748751897914513273944533

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