Properties

Label 2-570-285.29-c1-0-2
Degree $2$
Conductor $570$
Sign $-0.967 - 0.253i$
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  + (−0.984 + 0.173i)2-s + (1.67 + 0.441i)3-s + (0.939 − 0.342i)4-s + (−1.90 − 1.16i)5-s + (−1.72 − 0.144i)6-s + (−2.42 − 1.40i)7-s + (−0.866 + 0.5i)8-s + (2.60 + 1.48i)9-s + (2.08 + 0.819i)10-s + (−5.08 + 2.93i)11-s + (1.72 − 0.157i)12-s + (−2.44 − 2.05i)13-s + (2.63 + 0.959i)14-s + (−2.67 − 2.79i)15-s + (0.766 − 0.642i)16-s + (0.740 + 4.20i)17-s + ⋯
L(s)  = 1  + (−0.696 + 0.122i)2-s + (0.966 + 0.255i)3-s + (0.469 − 0.171i)4-s + (−0.852 − 0.522i)5-s + (−0.704 − 0.0589i)6-s + (−0.918 − 0.530i)7-s + (−0.306 + 0.176i)8-s + (0.869 + 0.493i)9-s + (0.657 + 0.259i)10-s + (−1.53 + 0.884i)11-s + (0.497 − 0.0454i)12-s + (−0.678 − 0.569i)13-s + (0.704 + 0.256i)14-s + (−0.691 − 0.722i)15-s + (0.191 − 0.160i)16-s + (0.179 + 1.01i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.967 - 0.253i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.967 - 0.253i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.967 - 0.253i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ -0.967 - 0.253i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0265369 + 0.206206i\)
\(L(\frac12)\) \(\approx\) \(0.0265369 + 0.206206i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.984 - 0.173i)T \)
3 \( 1 + (-1.67 - 0.441i)T \)
5 \( 1 + (1.90 + 1.16i)T \)
19 \( 1 + (1.47 - 4.10i)T \)
good7 \( 1 + (2.42 + 1.40i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (5.08 - 2.93i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.44 + 2.05i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (-0.740 - 4.20i)T + (-15.9 + 5.81i)T^{2} \)
23 \( 1 + (7.84 - 2.85i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (-0.140 + 0.794i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (0.327 + 0.189i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 0.577T + 37T^{2} \)
41 \( 1 + (-6.13 + 5.14i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-0.0911 + 0.250i)T + (-32.9 - 27.6i)T^{2} \)
47 \( 1 + (-1.53 + 8.68i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (-1.39 - 3.84i)T + (-40.6 + 34.0i)T^{2} \)
59 \( 1 + (1.97 + 11.2i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (5.56 - 2.02i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-0.0547 + 0.310i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (11.7 + 4.29i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (0.930 + 1.10i)T + (-12.6 + 71.8i)T^{2} \)
79 \( 1 + (-0.149 - 0.178i)T + (-13.7 + 77.7i)T^{2} \)
83 \( 1 + (1.81 - 3.15i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (11.3 + 9.50i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (-2.19 - 12.4i)T + (-91.1 + 33.1i)T^{2} \)
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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

−10.55846167471776920697084329575, −10.19069239932313842971889888049, −9.476821306886798642255769534357, −8.250521820772733231284762624744, −7.83381007577314726686072768934, −7.18723047314755178737595399847, −5.65676382408642375343931595403, −4.32475856416333309974597579558, −3.39509086626318826877439531073, −2.06546362553260856214285594997, 0.11973530885247993523862602081, 2.67986639580866039430435698841, 2.85648157764595498643303704705, 4.37101309350045477559156752467, 6.07358250193529846721869453871, 7.09514562625993481401660675326, 7.77191970843025353159737549648, 8.522426741335626551069198058567, 9.400338103261589052785905522508, 10.15340663450173151553935558574

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