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Properties

Label 2-57-57.8-c1-0-2
Degree $2$
Conductor $57$
Sign $0.736 + 0.676i$
Analytic cond. $0.455147$
Root an. cond. $0.674646$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.965 − 1.67i)2-s + (0.158 + 1.72i)3-s + (−0.866 − 1.50i)4-s + (−1.22 − 0.707i)5-s + (3.03 + 1.40i)6-s − 3.73·7-s + 0.517·8-s + (−2.94 + 0.548i)9-s + (−2.36 + 1.36i)10-s + 0.378i·11-s + (2.44 − 1.73i)12-s + (3.23 − 1.86i)13-s + (−3.60 + 6.24i)14-s + (1.02 − 2.22i)15-s + (2.23 − 3.86i)16-s + (4.24 + 2.44i)17-s + ⋯
L(s)  = 1  + (0.683 − 1.18i)2-s + (0.0917 + 0.995i)3-s + (−0.433 − 0.750i)4-s + (−0.547 − 0.316i)5-s + (1.24 + 0.571i)6-s − 1.41·7-s + 0.183·8-s + (−0.983 + 0.182i)9-s + (−0.748 + 0.431i)10-s + 0.114i·11-s + (0.707 − 0.499i)12-s + (0.896 − 0.517i)13-s + (−0.963 + 1.66i)14-s + (0.264 − 0.574i)15-s + (0.558 − 0.966i)16-s + (1.02 + 0.594i)17-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(57\)    =    \(3 \cdot 19\)
Sign: $0.736 + 0.676i$
Analytic conductor: \(0.455147\)
Root analytic conductor: \(0.674646\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{57} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 57,\ (\ :1/2),\ 0.736 + 0.676i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.989391 - 0.385150i\)
\(L(\frac12)\) \(\approx\) \(0.989391 - 0.385150i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.158 - 1.72i)T \)
19 \( 1 + (1.73 - 4i)T \)
good2 \( 1 + (-0.965 + 1.67i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (1.22 + 0.707i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + 3.73T + 7T^{2} \)
11 \( 1 - 0.378iT - 11T^{2} \)
13 \( 1 + (-3.23 + 1.86i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-4.24 - 2.44i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (-0.328 + 0.189i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.86 + 6.69i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.46iT - 31T^{2} \)
37 \( 1 - 4.26iT - 37T^{2} \)
41 \( 1 + (2.82 - 4.89i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.13 - 1.96i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-9.14 + 5.27i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.01 + 5.22i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.19 - 7.26i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.76 - 3.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.866 + 0.5i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.79 - 3.10i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.5 + 2.59i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.06 + 1.76i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 7.72iT - 83T^{2} \)
89 \( 1 + (3.67 + 6.36i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.46 - 3.73i)T + (48.5 + 84.0i)T^{2} \)
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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

−15.07854053589529969808798871629, −13.71150526939838952570908991866, −12.68967162030574230732514640321, −11.79532622624805397589616112208, −10.48634358868941295567336056076, −9.822361706184052610501883162566, −8.210898813423859938003572042116, −5.83538560012830027510054089031, −4.07364912544682972983365394053, −3.24585034748799857323106711197, 3.50535950209546512577027185642, 5.74691025437395243562210036600, 6.77308743331003609301498964328, 7.53804837980126129050672743526, 9.076620864366065152782576695960, 11.05371291280800259296467727394, 12.49823207055216922092088648722, 13.36810442706286026438303955763, 14.19131390497935024582520341465, 15.39162529626447705955556754895

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