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Properties

Label 2-570-5.4-c1-0-13
Degree $2$
Conductor $570$
Sign $0.139 + 0.990i$
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 + (0.311 + 2.21i)5-s + 6-s − 4.42i·7-s − i·8-s − 9-s + (−2.21 + 0.311i)10-s − 5.80·11-s + i·12-s − 6.42i·13-s + 4.42·14-s + (2.21 − 0.311i)15-s + 16-s + 3.37i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + (0.139 + 0.990i)5-s + 0.408·6-s − 1.67i·7-s − 0.353i·8-s − 0.333·9-s + (−0.700 + 0.0983i)10-s − 1.75·11-s + 0.288i·12-s − 1.78i·13-s + 1.18·14-s + (0.571 − 0.0803i)15-s + 0.250·16-s + 0.819i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.139 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.139 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.139 + 0.990i$
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: \((2,\ 570,\ (\ :1/2),\ 0.139 + 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.648412 - 0.563680i\)
\(L(\frac12)\) \(\approx\) \(0.648412 - 0.563680i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.311 - 2.21i)T \)
19 \( 1 - T \)
good7 \( 1 + 4.42iT - 7T^{2} \)
11 \( 1 + 5.80T + 11T^{2} \)
13 \( 1 + 6.42iT - 13T^{2} \)
17 \( 1 - 3.37iT - 17T^{2} \)
23 \( 1 + 6.42iT - 23T^{2} \)
29 \( 1 + 7.80T + 29T^{2} \)
31 \( 1 - 9.05T + 31T^{2} \)
37 \( 1 + 3.67iT - 37T^{2} \)
41 \( 1 - 4.42T + 41T^{2} \)
43 \( 1 + 1.05iT - 43T^{2} \)
47 \( 1 + 5.18iT - 47T^{2} \)
53 \( 1 + 4.75iT - 53T^{2} \)
59 \( 1 + 4.62T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 + 2.75iT - 67T^{2} \)
71 \( 1 - 7.61T + 71T^{2} \)
73 \( 1 - 11.6iT - 73T^{2} \)
79 \( 1 + 2.94T + 79T^{2} \)
83 \( 1 + 0.133iT - 83T^{2} \)
89 \( 1 - 3.18T + 89T^{2} \)
97 \( 1 - 11.4iT - 97T^{2} \)
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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

−10.46920883998416672989474510015, −10.02073803068068373526703433595, −8.150973355020046924761385755336, −7.80285425614020946423791794270, −7.08521448854715984076807005390, −6.13810873167433706422579095714, −5.19134533922343087387048329086, −3.77470426840634419183457133338, −2.65442586363393421800977762663, −0.46886071688508549989339828443, 1.94944813339847646432386961427, 2.92548584134605572906950871301, 4.49345870289209515427100951387, 5.19031312895059595666947136059, 5.91022455048284462226917151535, 7.72650257220208633099541566250, 8.658317778478065352600893451696, 9.394464615467742859905946365946, 9.729449641050596163540269287107, 11.15719796020929095425533780850

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