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Properties

Label 2-570-95.37-c1-0-10
Degree $2$
Conductor $570$
Sign $0.699 - 0.714i$
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 + (1.42 − 1.72i)5-s − 1.00·6-s + (3.40 + 3.40i)7-s + (0.707 + 0.707i)8-s + 1.00i·9-s + (0.215 + 2.22i)10-s + 3.94·11-s + (0.707 − 0.707i)12-s + (−4.03 − 4.03i)13-s − 4.81·14-s + (2.22 − 0.215i)15-s − 1.00·16-s + (−3.90 − 3.90i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (0.408 + 0.408i)3-s − 0.500i·4-s + (0.635 − 0.772i)5-s − 0.408·6-s + (1.28 + 1.28i)7-s + (0.250 + 0.250i)8-s + 0.333i·9-s + (0.0682 + 0.703i)10-s + 1.18·11-s + (0.204 − 0.204i)12-s + (−1.11 − 1.11i)13-s − 1.28·14-s + (0.574 − 0.0557i)15-s − 0.250·16-s + (−0.947 − 0.947i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.699 - 0.714i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.699 - 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.699 - 0.714i$
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: \((2,\ 570,\ (\ :1/2),\ 0.699 - 0.714i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.53064 + 0.643621i\)
\(L(\frac12)\) \(\approx\) \(1.53064 + 0.643621i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.42 + 1.72i)T \)
19 \( 1 + (-3.49 - 2.60i)T \)
good7 \( 1 + (-3.40 - 3.40i)T + 7iT^{2} \)
11 \( 1 - 3.94T + 11T^{2} \)
13 \( 1 + (4.03 + 4.03i)T + 13iT^{2} \)
17 \( 1 + (3.90 + 3.90i)T + 17iT^{2} \)
23 \( 1 + (-0.537 + 0.537i)T - 23iT^{2} \)
29 \( 1 - 6.28T + 29T^{2} \)
31 \( 1 - 7.00iT - 31T^{2} \)
37 \( 1 + (1.05 - 1.05i)T - 37iT^{2} \)
41 \( 1 + 3.03iT - 41T^{2} \)
43 \( 1 + (5.70 - 5.70i)T - 43iT^{2} \)
47 \( 1 + (2.04 + 2.04i)T + 47iT^{2} \)
53 \( 1 + (-4.39 - 4.39i)T + 53iT^{2} \)
59 \( 1 + 2.32T + 59T^{2} \)
61 \( 1 + 9.32T + 61T^{2} \)
67 \( 1 + (7.35 - 7.35i)T - 67iT^{2} \)
71 \( 1 + 9.62iT - 71T^{2} \)
73 \( 1 + (-4.47 + 4.47i)T - 73iT^{2} \)
79 \( 1 - 0.991T + 79T^{2} \)
83 \( 1 + (-6.64 + 6.64i)T - 83iT^{2} \)
89 \( 1 - 7.09T + 89T^{2} \)
97 \( 1 + (-7.11 + 7.11i)T - 97iT^{2} \)
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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

−10.58913074884041556498739535901, −9.644776175736605597938785874503, −8.976618029693136364035166893318, −8.465895493162639008130391582713, −7.54563239087487255892184981827, −6.21218434710826475884525913534, −5.08472177440466341023293502803, −4.81438998352313780866896472384, −2.71511501202018029244454072387, −1.49295005579740044072137870860, 1.40220863862560393811927972512, 2.27443804724121211961452382716, 3.80833102360133585045751923261, 4.70149144151811684440347911526, 6.57022205418313128692734602158, 7.07670856995599220142036508782, 7.936955764177043119870781593443, 9.017409889680221581565547047923, 9.755733850937146047161299569652, 10.64784726247105894191298011999

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