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Properties

Label 2-570-95.37-c1-0-2
Degree $2$
Conductor $570$
Sign $-0.246 - 0.969i$
Analytic cond. $4.55147$
Root an. cond. $2.13341$
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  + (−0.707 + 0.707i)2-s + (−0.707 − 0.707i)3-s − 1.00i·4-s + (1.25 + 1.84i)5-s + 1.00·6-s + (3.10 + 3.10i)7-s + (0.707 + 0.707i)8-s + 1.00i·9-s + (−2.19 − 0.416i)10-s − 3.82·11-s + (−0.707 + 0.707i)12-s + (0.0891 + 0.0891i)13-s − 4.39·14-s + (0.416 − 2.19i)15-s − 1.00·16-s + (−1.83 − 1.83i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (−0.408 − 0.408i)3-s − 0.500i·4-s + (0.562 + 0.826i)5-s + 0.408·6-s + (1.17 + 1.17i)7-s + (0.250 + 0.250i)8-s + 0.333i·9-s + (−0.694 − 0.131i)10-s − 1.15·11-s + (−0.204 + 0.204i)12-s + (0.0247 + 0.0247i)13-s − 1.17·14-s + (0.107 − 0.567i)15-s − 0.250·16-s + (−0.443 − 0.443i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.246 - 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.246 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.246 - 0.969i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ -0.246 - 0.969i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.638727 + 0.821928i\)
\(L(\frac12)\) \(\approx\) \(0.638727 + 0.821928i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 + (-1.25 - 1.84i)T \)
19 \( 1 + (-2.70 - 3.41i)T \)
good7 \( 1 + (-3.10 - 3.10i)T + 7iT^{2} \)
11 \( 1 + 3.82T + 11T^{2} \)
13 \( 1 + (-0.0891 - 0.0891i)T + 13iT^{2} \)
17 \( 1 + (1.83 + 1.83i)T + 17iT^{2} \)
23 \( 1 + (-2.58 + 2.58i)T - 23iT^{2} \)
29 \( 1 + 3.60T + 29T^{2} \)
31 \( 1 - 3.60iT - 31T^{2} \)
37 \( 1 + (7.07 - 7.07i)T - 37iT^{2} \)
41 \( 1 - 11.2iT - 41T^{2} \)
43 \( 1 + (-7.93 + 7.93i)T - 43iT^{2} \)
47 \( 1 + (0.463 + 0.463i)T + 47iT^{2} \)
53 \( 1 + (-3.21 - 3.21i)T + 53iT^{2} \)
59 \( 1 + 9.40T + 59T^{2} \)
61 \( 1 - 8.21T + 61T^{2} \)
67 \( 1 + (-8.78 + 8.78i)T - 67iT^{2} \)
71 \( 1 - 1.66iT - 71T^{2} \)
73 \( 1 + (3.64 - 3.64i)T - 73iT^{2} \)
79 \( 1 - 8.82T + 79T^{2} \)
83 \( 1 + (0.347 - 0.347i)T - 83iT^{2} \)
89 \( 1 - 9.79T + 89T^{2} \)
97 \( 1 + (-6.88 + 6.88i)T - 97iT^{2} \)
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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

−10.92981239509854817959475677796, −10.19696391939890559357741996814, −9.121119993164190139881297303902, −8.204927305019591687861892808648, −7.47995737000204540598714139102, −6.49481450744247386481705909123, −5.51700443116705209526287353771, −5.02308072944675261996671990277, −2.76874744480062839816490167832, −1.75237792292057775905766191176, 0.75226089350219586088090635219, 2.10522909792603599699057653833, 3.87576823019195466463100181018, 4.84376097947893098641149731169, 5.54713633429641002168559770229, 7.17882599404248973966299517050, 7.914302578605021857994860910597, 8.895165356637791148957873356964, 9.688212807118013359411148065767, 10.79108207513226437289058311354

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