Properties

Label 2-570-5.4-c1-0-1
Degree $2$
Conductor $570$
Sign $-0.749 + 0.662i$
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  + i·2-s + i·3-s − 4-s + (−1.67 + 1.48i)5-s − 6-s + 3.35i·7-s i·8-s − 9-s + (−1.48 − 1.67i)10-s − 0.962·11-s i·12-s − 1.61i·13-s − 3.35·14-s + (−1.48 − 1.67i)15-s + 16-s + 0.387i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.577i·3-s − 0.5·4-s + (−0.749 + 0.662i)5-s − 0.408·6-s + 1.26i·7-s − 0.353i·8-s − 0.333·9-s + (−0.468 − 0.529i)10-s − 0.290·11-s − 0.288i·12-s − 0.447i·13-s − 0.895·14-s + (−0.382 − 0.432i)15-s + 0.250·16-s + 0.0940i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 + 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.749 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.749 + 0.662i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ -0.749 + 0.662i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.242044 - 0.639135i\)
\(L(\frac12)\) \(\approx\) \(0.242044 - 0.639135i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 - iT \)
5 \( 1 + (1.67 - 1.48i)T \)
19 \( 1 + T \)
good7 \( 1 - 3.35iT - 7T^{2} \)
11 \( 1 + 0.962T + 11T^{2} \)
13 \( 1 + 1.61iT - 13T^{2} \)
17 \( 1 - 0.387iT - 17T^{2} \)
23 \( 1 + 0.962iT - 23T^{2} \)
29 \( 1 + 6.96T + 29T^{2} \)
31 \( 1 - 3.35T + 31T^{2} \)
37 \( 1 - 1.61iT - 37T^{2} \)
41 \( 1 + 9.27T + 41T^{2} \)
43 \( 1 - 6.18iT - 43T^{2} \)
47 \( 1 + 0.962iT - 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 + 10.3T + 59T^{2} \)
61 \( 1 - 11.9T + 61T^{2} \)
67 \( 1 - 7.22iT - 67T^{2} \)
71 \( 1 - 7.22T + 71T^{2} \)
73 \( 1 + 3.22iT - 73T^{2} \)
79 \( 1 - 3.35T + 79T^{2} \)
83 \( 1 - 15.0iT - 83T^{2} \)
89 \( 1 + 4.64T + 89T^{2} \)
97 \( 1 - 10.9iT - 97T^{2} \)
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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

−11.23811577132475655377859608723, −10.33717637263800498103372671748, −9.411259596512340013290592119127, −8.482366464785332251064297780928, −7.87201827409224188344203101939, −6.71644791570147030826565811817, −5.79392936258933716759737369588, −4.91069135163245847719790275573, −3.71221800076445773810352542758, −2.62905359638996164785532931403, 0.38826299121997028302726111760, 1.73330909398726082051822632408, 3.43705856575655104629233546405, 4.27377215473536714194381455765, 5.30428127178339091425732015310, 6.80996599760350786897894865851, 7.61454655213406547320648486144, 8.390777369358195714726312157530, 9.358410936148320390042214983159, 10.36937190526628826602065150412

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