Properties

Label 2-570-95.8-c1-0-8
Degree $2$
Conductor $570$
Sign $0.999 - 0.0215i$
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  + (0.965 − 0.258i)2-s + (0.258 + 0.965i)3-s + (0.866 − 0.499i)4-s + (−1.99 − 1.01i)5-s + (0.499 + 0.866i)6-s + (−0.219 + 0.219i)7-s + (0.707 − 0.707i)8-s + (−0.866 + 0.499i)9-s + (−2.18 − 0.465i)10-s + 6.33·11-s + (0.707 + 0.707i)12-s + (3.59 + 0.962i)13-s + (−0.155 + 0.268i)14-s + (0.465 − 2.18i)15-s + (0.500 − 0.866i)16-s + (0.885 + 3.30i)17-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (0.149 + 0.557i)3-s + (0.433 − 0.249i)4-s + (−0.890 − 0.454i)5-s + (0.204 + 0.353i)6-s + (−0.0828 + 0.0828i)7-s + (0.249 − 0.249i)8-s + (−0.288 + 0.166i)9-s + (−0.691 − 0.147i)10-s + 1.90·11-s + (0.204 + 0.204i)12-s + (0.996 + 0.267i)13-s + (−0.0414 + 0.0717i)14-s + (0.120 − 0.564i)15-s + (0.125 − 0.216i)16-s + (0.214 + 0.801i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0215i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0215i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.999 - 0.0215i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.999 - 0.0215i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.21262 + 0.0237916i\)
\(L(\frac12)\) \(\approx\) \(2.21262 + 0.0237916i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 + 0.258i)T \)
3 \( 1 + (-0.258 - 0.965i)T \)
5 \( 1 + (1.99 + 1.01i)T \)
19 \( 1 + (-3.97 + 1.79i)T \)
good7 \( 1 + (0.219 - 0.219i)T - 7iT^{2} \)
11 \( 1 - 6.33T + 11T^{2} \)
13 \( 1 + (-3.59 - 0.962i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (-0.885 - 3.30i)T + (-14.7 + 8.5i)T^{2} \)
23 \( 1 + (-0.601 + 2.24i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (3.42 + 5.93i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 2.71iT - 31T^{2} \)
37 \( 1 + (-0.817 - 0.817i)T + 37iT^{2} \)
41 \( 1 + (8.81 + 5.08i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (8.44 - 2.26i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (10.1 + 2.73i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-3.96 - 1.06i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-0.367 + 0.636i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.17 - 3.76i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.33 - 4.98i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (1.14 + 0.663i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-4.65 + 1.24i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-7.21 + 12.5i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.190 + 0.190i)T + 83iT^{2} \)
89 \( 1 + (-5.18 - 8.97i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (15.9 - 4.26i)T + (84.0 - 48.5i)T^{2} \)
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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

−11.05932787723645112520826529536, −9.862726958058077351010228148813, −8.969832350996579293132784732308, −8.270058412482333830381923052055, −6.96013180956637083567624094106, −6.09776482671504660401380702241, −4.87225841808326873245531365409, −3.91612016688123667737156674441, −3.43842845463303012714149577929, −1.41162700999618990440542151772, 1.39540547128346122896817246017, 3.32961324162913203876089498862, 3.72590157040067348262454303205, 5.17095262211313942714408155259, 6.45500086714140129043792864879, 6.92752442073791454949781812775, 7.87803958567129329610836419364, 8.770700307079163507536572363513, 9.846983250117558172657455053154, 11.31510486565505405617873878328

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