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Properties

Label 2-570-95.27-c1-0-18
Degree $2$
Conductor $570$
Sign $-0.518 + 0.854i$
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  + (0.258 + 0.965i)2-s + (0.965 − 0.258i)3-s + (−0.866 + 0.499i)4-s + (−2.17 + 0.507i)5-s + (0.499 + 0.866i)6-s + (−2.55 − 2.55i)7-s + (−0.707 − 0.707i)8-s + (0.866 − 0.499i)9-s + (−1.05 − 1.97i)10-s − 2.10·11-s + (−0.707 + 0.707i)12-s + (−0.0901 + 0.336i)13-s + (1.80 − 3.12i)14-s + (−1.97 + 1.05i)15-s + (0.500 − 0.866i)16-s + (−4.16 + 1.11i)17-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s + (0.557 − 0.149i)3-s + (−0.433 + 0.249i)4-s + (−0.973 + 0.227i)5-s + (0.204 + 0.353i)6-s + (−0.965 − 0.965i)7-s + (−0.249 − 0.249i)8-s + (0.288 − 0.166i)9-s + (−0.333 − 0.623i)10-s − 0.633·11-s + (−0.204 + 0.204i)12-s + (−0.0250 + 0.0933i)13-s + (0.482 − 0.835i)14-s + (−0.509 + 0.272i)15-s + (0.125 − 0.216i)16-s + (−1.01 + 0.270i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.518 + 0.854i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.518 + 0.854i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.518 + 0.854i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ -0.518 + 0.854i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.133472 - 0.237107i\)
\(L(\frac12)\) \(\approx\) \(0.133472 - 0.237107i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 - 0.965i)T \)
3 \( 1 + (-0.965 + 0.258i)T \)
5 \( 1 + (2.17 - 0.507i)T \)
19 \( 1 + (4.05 + 1.61i)T \)
good7 \( 1 + (2.55 + 2.55i)T + 7iT^{2} \)
11 \( 1 + 2.10T + 11T^{2} \)
13 \( 1 + (0.0901 - 0.336i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + (4.16 - 1.11i)T + (14.7 - 8.5i)T^{2} \)
23 \( 1 + (3.42 + 0.916i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (-0.205 - 0.355i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 9.24iT - 31T^{2} \)
37 \( 1 + (3.61 - 3.61i)T - 37iT^{2} \)
41 \( 1 + (-6.95 - 4.01i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.601 - 2.24i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (1.74 - 6.52i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-3.33 + 12.4i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (6.55 - 11.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.171 - 0.296i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.76 + 1.54i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-0.442 - 0.255i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.48 - 5.52i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-4.99 + 8.64i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.69 - 4.69i)T - 83iT^{2} \)
89 \( 1 + (-8.24 - 14.2i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.35 + 12.5i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.38715747761880906445939157462, −9.438911077752028832783372326633, −8.389920604620477895017215136758, −7.72147796912184377337149882766, −6.90110065167777738705443818681, −6.21438873980316215449256220131, −4.47585618895094683615430690985, −3.90599658755402064208686794301, −2.70942320580022787227566591877, −0.12738461348684766349780192535, 2.24500789290042291981598809558, 3.22176790659450777591950641140, 4.16694090692398796192710853233, 5.25777745429347046917856648166, 6.48031536254916655284102784015, 7.68122621007794994236200164110, 8.735790186980698683916567741003, 9.090380632471800056841793365489, 10.28505318654795177147183809660, 10.95200535184517405009565115137

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