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Properties

Label 2-520-104.69-c1-0-53
Degree $2$
Conductor $520$
Sign $-0.879 + 0.475i$
Analytic cond. $4.15222$
Root an. cond. $2.03769$
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  + (1.23 − 0.684i)2-s + (−1.48 − 0.858i)3-s + (1.06 − 1.69i)4-s + 5-s + (−2.42 − 0.0445i)6-s + (−3.27 + 1.89i)7-s + (0.155 − 2.82i)8-s + (−0.0260 − 0.0450i)9-s + (1.23 − 0.684i)10-s + (1.16 − 2.01i)11-s + (−3.03 + 1.60i)12-s + (−0.0528 − 3.60i)13-s + (−2.75 + 4.58i)14-s + (−1.48 − 0.858i)15-s + (−1.74 − 3.60i)16-s + (−3.14 − 5.44i)17-s + ⋯
L(s)  = 1  + (0.875 − 0.484i)2-s + (−0.858 − 0.495i)3-s + (0.531 − 0.847i)4-s + 0.447·5-s + (−0.991 − 0.0182i)6-s + (−1.23 + 0.714i)7-s + (0.0550 − 0.998i)8-s + (−0.00867 − 0.0150i)9-s + (0.391 − 0.216i)10-s + (0.351 − 0.608i)11-s + (−0.876 + 0.463i)12-s + (−0.0146 − 0.999i)13-s + (−0.737 + 1.22i)14-s + (−0.383 − 0.221i)15-s + (−0.435 − 0.900i)16-s + (−0.763 − 1.32i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.879 + 0.475i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.879 + 0.475i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(520\)    =    \(2^{3} \cdot 5 \cdot 13\)
Sign: $-0.879 + 0.475i$
Analytic conductor: \(4.15222\)
Root analytic conductor: \(2.03769\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{520} (381, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 520,\ (\ :1/2),\ -0.879 + 0.475i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.335951 - 1.32676i\)
\(L(\frac12)\) \(\approx\) \(0.335951 - 1.32676i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.23 + 0.684i)T \)
5 \( 1 - T \)
13 \( 1 + (0.0528 + 3.60i)T \)
good3 \( 1 + (1.48 + 0.858i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + (3.27 - 1.89i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.16 + 2.01i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (3.14 + 5.44i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.665 + 1.15i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.99 - 5.18i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.31 - 0.758i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 0.175iT - 31T^{2} \)
37 \( 1 + (-2.04 + 3.53i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-10.9 - 6.31i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-7.71 + 4.45i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 3.15iT - 47T^{2} \)
53 \( 1 + 0.254iT - 53T^{2} \)
59 \( 1 + (5.61 + 9.72i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.168 + 0.0972i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.63 + 4.55i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-11.0 + 6.36i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 1.93iT - 73T^{2} \)
79 \( 1 - 6.68T + 79T^{2} \)
83 \( 1 + 6.27T + 83T^{2} \)
89 \( 1 + (9.69 + 5.59i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (10.4 - 6.01i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.89976120060278524918975995381, −9.644004166606039300225152359478, −9.171604607368882908402569886612, −7.35983982339325874464795754120, −6.25016684834776358766671411642, −6.00033110919669946346762068037, −5.05657712360313135332348301535, −3.45099565030957852296423184331, −2.52326247952343878936483304640, −0.64568847587507841722283481101, 2.34760249769605589570304541796, 4.05590418130357855803532800119, 4.42627221044200414198726524448, 5.92458913699244833851513463988, 6.34154664420354679364991667038, 7.17283682217754247460372751901, 8.524605089221684284864119301996, 9.687109131853891705326556057920, 10.51817862387994483275626891383, 11.21360653612276568912871759411

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