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Properties

Label 2-57-57.50-c1-0-0
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  + (0.258 + 0.448i)2-s + (−0.158 + 1.72i)3-s + (0.866 − 1.5i)4-s + (−1.22 + 0.707i)5-s + (−0.814 + 0.375i)6-s − 0.267·7-s + 1.93·8-s + (−2.94 − 0.548i)9-s + (−0.633 − 0.366i)10-s − 5.27i·11-s + (2.44 + 1.73i)12-s + (−0.232 − 0.133i)13-s + (−0.0693 − 0.120i)14-s + (−1.02 − 2.22i)15-s + (−1.23 − 2.13i)16-s + (−4.24 + 2.44i)17-s + ⋯
L(s)  = 1  + (0.183 + 0.316i)2-s + (−0.0917 + 0.995i)3-s + (0.433 − 0.750i)4-s + (−0.547 + 0.316i)5-s + (−0.332 + 0.153i)6-s − 0.101·7-s + 0.683·8-s + (−0.983 − 0.182i)9-s + (−0.200 − 0.115i)10-s − 1.59i·11-s + (0.707 + 0.499i)12-s + (−0.0643 − 0.0371i)13-s + (−0.0185 − 0.0321i)14-s + (−0.264 − 0.574i)15-s + (−0.308 − 0.533i)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: $\chi_{57} (50, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 57,\ (\ :1/2),\ 0.736 - 0.676i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.849202 + 0.330578i\)
\(L(\frac12)\) \(\approx\) \(0.849202 + 0.330578i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.258 - 0.448i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (1.22 - 0.707i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + 0.267T + 7T^{2} \)
11 \( 1 + 5.27iT - 11T^{2} \)
13 \( 1 + (0.232 + 0.133i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (4.24 - 2.44i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (-4.57 - 2.63i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.03 - 1.79i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 2.46iT - 31T^{2} \)
37 \( 1 - 7.73iT - 37T^{2} \)
41 \( 1 + (-2.82 - 4.89i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.86 + 4.96i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.656 - 0.378i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.46 + 9.46i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.60 + 9.71i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.23 + 9.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.866 + 0.5i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-6.69 - 11.5i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.5 - 2.59i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-9.06 + 5.23i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 2.07iT - 83T^{2} \)
89 \( 1 + (3.67 - 6.36i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.464 - 0.267i)T + (48.5 - 84.0i)T^{2} \)
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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

−15.40050144011617437473108453065, −14.60317958847261954004999217725, −13.50046195624061398573741903240, −11.40165953683391948612273551694, −10.97573814793857824693443748029, −9.694546998156104682714728264106, −8.244163149286682343814067495788, −6.45412264235718107950849014603, −5.27793194511322645048940879153, −3.51104163532796405344468063448, 2.45490667272061577685300869323, 4.56780582559173079407586771913, 6.87053893996983245047121915717, 7.56668189090158669657624563160, 8.981943675111857193351138660895, 10.97239530362242185913511213800, 12.00200031759719286697900489594, 12.64758821509313320099169680561, 13.57921679508008633843632308007, 15.15309641161560415475659488155

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