Properties

Label 2-57-19.9-c1-0-2
Degree $2$
Conductor $57$
Sign $-0.527 + 0.849i$
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  + (−0.458 − 2.59i)2-s + (0.766 − 0.642i)3-s + (−4.65 + 1.69i)4-s + (1.91 + 0.698i)5-s + (−2.02 − 1.69i)6-s + (−1.30 + 2.26i)7-s + (3.89 + 6.75i)8-s + (0.173 − 0.984i)9-s + (0.934 − 5.30i)10-s + (−1.86 − 3.22i)11-s + (−2.47 + 4.29i)12-s + (1.65 + 1.38i)13-s + (6.48 + 2.36i)14-s + (1.91 − 0.698i)15-s + (8.16 − 6.85i)16-s + (−0.289 − 1.63i)17-s + ⋯
L(s)  = 1  + (−0.323 − 1.83i)2-s + (0.442 − 0.371i)3-s + (−2.32 + 0.847i)4-s + (0.857 + 0.312i)5-s + (−0.824 − 0.692i)6-s + (−0.494 + 0.856i)7-s + (1.37 + 2.38i)8-s + (0.0578 − 0.328i)9-s + (0.295 − 1.67i)10-s + (−0.561 − 0.971i)11-s + (−0.715 + 1.23i)12-s + (0.459 + 0.385i)13-s + (1.73 + 0.630i)14-s + (0.495 − 0.180i)15-s + (2.04 − 1.71i)16-s + (−0.0701 − 0.397i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.527 + 0.849i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.527 + 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(57\)    =    \(3 \cdot 19\)
Sign: $-0.527 + 0.849i$
Analytic conductor: \(0.455147\)
Root analytic conductor: \(0.674646\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{57} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 57,\ (\ :1/2),\ -0.527 + 0.849i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.386285 - 0.694514i\)
\(L(\frac12)\) \(\approx\) \(0.386285 - 0.694514i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.766 + 0.642i)T \)
19 \( 1 + (-1.99 - 3.87i)T \)
good2 \( 1 + (0.458 + 2.59i)T + (-1.87 + 0.684i)T^{2} \)
5 \( 1 + (-1.91 - 0.698i)T + (3.83 + 3.21i)T^{2} \)
7 \( 1 + (1.30 - 2.26i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.86 + 3.22i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.65 - 1.38i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (0.289 + 1.63i)T + (-15.9 + 5.81i)T^{2} \)
23 \( 1 + (6.08 - 2.21i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (0.591 - 3.35i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (-0.931 + 1.61i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 0.727T + 37T^{2} \)
41 \( 1 + (-4.86 + 4.07i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (6.83 + 2.48i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-0.262 + 1.48i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (1.29 - 0.471i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (1.27 + 7.25i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-10.5 + 3.84i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-1.15 + 6.57i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (5.54 + 2.01i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (10.5 - 8.81i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (-10.2 + 8.58i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (4.34 - 7.52i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (5.47 + 4.59i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (-0.402 - 2.28i)T + (-91.1 + 33.1i)T^{2} \)
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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

−14.11559149188218864751715190776, −13.52938726191547706501718681396, −12.46806122045929821715878977363, −11.47243874502859490989895486333, −10.18520938385769718374562690750, −9.328152025995528592252112191802, −8.266158085116263558239115662079, −5.81182121331692989258215221161, −3.40305482538286574116304696317, −2.12277977908158716285408541930, 4.44008446461088685440834477548, 5.82996782801612562649673294084, 7.13978614533722375295476123913, 8.280547210536957399646030378044, 9.585397437720370858587169394579, 10.18903176742122358403885079052, 13.12196693350862031036109204315, 13.63104808958661324766995139223, 14.76085857264128274036426397562, 15.72764241823696250517527167158

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