Properties

Label 2-570-19.6-c1-0-2
Degree $2$
Conductor $570$
Sign $-0.320 - 0.947i$
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  + (−0.766 + 0.642i)2-s + (0.939 + 0.342i)3-s + (0.173 − 0.984i)4-s + (0.173 + 0.984i)5-s + (−0.939 + 0.342i)6-s + (−0.266 + 0.460i)7-s + (0.500 + 0.866i)8-s + (0.766 + 0.642i)9-s + (−0.766 − 0.642i)10-s + (1.67 + 2.89i)11-s + (0.499 − 0.866i)12-s + (−5.19 + 1.89i)13-s + (−0.0923 − 0.524i)14-s + (−0.173 + 0.984i)15-s + (−0.939 − 0.342i)16-s + (3.50 − 2.94i)17-s + ⋯
L(s)  = 1  + (−0.541 + 0.454i)2-s + (0.542 + 0.197i)3-s + (0.0868 − 0.492i)4-s + (0.0776 + 0.440i)5-s + (−0.383 + 0.139i)6-s + (−0.100 + 0.174i)7-s + (0.176 + 0.306i)8-s + (0.255 + 0.214i)9-s + (−0.242 − 0.203i)10-s + (0.504 + 0.874i)11-s + (0.144 − 0.250i)12-s + (−1.44 + 0.524i)13-s + (−0.0246 − 0.140i)14-s + (−0.0448 + 0.254i)15-s + (−0.234 − 0.0855i)16-s + (0.849 − 0.713i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.320 - 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.320 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.320 - 0.947i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (481, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ -0.320 - 0.947i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.708724 + 0.988294i\)
\(L(\frac12)\) \(\approx\) \(0.708724 + 0.988294i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.766 - 0.642i)T \)
3 \( 1 + (-0.939 - 0.342i)T \)
5 \( 1 + (-0.173 - 0.984i)T \)
19 \( 1 + (0.0320 - 4.35i)T \)
good7 \( 1 + (0.266 - 0.460i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.67 - 2.89i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (5.19 - 1.89i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (-3.50 + 2.94i)T + (2.95 - 16.7i)T^{2} \)
23 \( 1 + (-0.0923 + 0.524i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (-6.36 - 5.34i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (3.93 - 6.81i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 5.02T + 37T^{2} \)
41 \( 1 + (10.2 + 3.72i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (1.66 + 9.42i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (-4.28 - 3.59i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 + (0.961 - 5.45i)T + (-49.8 - 18.1i)T^{2} \)
59 \( 1 + (-9.57 + 8.03i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (0.432 - 2.45i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (3.75 + 3.15i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (-0.936 - 5.30i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (14.1 + 5.14i)T + (55.9 + 46.9i)T^{2} \)
79 \( 1 + (-7.90 - 2.87i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (-1.66 + 2.88i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (9.13 - 3.32i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (5.52 - 4.63i)T + (16.8 - 95.5i)T^{2} \)
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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

−10.58333333906619681991853536164, −9.939926083744990806482667486586, −9.333231656706248268052428544965, −8.404388408605609949981691477767, −7.23849824183718882574912449182, −6.97024895265561821295330782224, −5.51413733043928893390290358428, −4.51196378798737725531282555328, −3.08106771429625765471144795738, −1.81992389596498091830316286085, 0.794550516355902460409054775725, 2.36644933394528704446493398275, 3.43489908611881342465906104312, 4.64938655325479486607847845225, 5.97092740667559132128386121448, 7.16165292500720308913561494010, 8.031953938006942946574014611891, 8.693430006916073880899288412176, 9.711345269133075318953096550464, 10.15277429187252400919689300939

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