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Properties

Label 2-570-95.27-c1-0-2
Degree $2$
Conductor $570$
Sign $0.366 - 0.930i$
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 + (1.20 + 1.88i)5-s + (0.499 + 0.866i)6-s + (−0.368 − 0.368i)7-s + (0.707 + 0.707i)8-s + (0.866 − 0.499i)9-s + (1.51 − 1.64i)10-s + 0.0597·11-s + (0.707 − 0.707i)12-s + (−0.198 + 0.741i)13-s + (−0.260 + 0.451i)14-s + (−1.64 − 1.51i)15-s + (0.500 − 0.866i)16-s + (−7.07 + 1.89i)17-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s + (−0.557 + 0.149i)3-s + (−0.433 + 0.249i)4-s + (0.537 + 0.843i)5-s + (0.204 + 0.353i)6-s + (−0.139 − 0.139i)7-s + (0.249 + 0.249i)8-s + (0.288 − 0.166i)9-s + (0.477 − 0.521i)10-s + 0.0180·11-s + (0.204 − 0.204i)12-s + (−0.0550 + 0.205i)13-s + (−0.0696 + 0.120i)14-s + (−0.425 − 0.389i)15-s + (0.125 − 0.216i)16-s + (−1.71 + 0.459i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.366 - 0.930i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.366 - 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.366 - 0.930i$
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: \((2,\ 570,\ (\ :1/2),\ 0.366 - 0.930i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.697012 + 0.474543i\)
\(L(\frac12)\) \(\approx\) \(0.697012 + 0.474543i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.20 - 1.88i)T \)
19 \( 1 + (2.26 - 3.72i)T \)
good7 \( 1 + (0.368 + 0.368i)T + 7iT^{2} \)
11 \( 1 - 0.0597T + 11T^{2} \)
13 \( 1 + (0.198 - 0.741i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + (7.07 - 1.89i)T + (14.7 - 8.5i)T^{2} \)
23 \( 1 + (-6.38 - 1.71i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (-2.71 - 4.70i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 3.60iT - 31T^{2} \)
37 \( 1 + (6.95 - 6.95i)T - 37iT^{2} \)
41 \( 1 + (-10.3 - 5.95i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.66 + 9.94i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (-1.02 + 3.81i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (3.27 - 12.2i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-0.464 + 0.804i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.593 - 1.02i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (11.4 + 3.06i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-7.61 - 4.39i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-3.66 - 13.6i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (1.71 - 2.97i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-8.15 + 8.15i)T - 83iT^{2} \)
89 \( 1 + (-1.67 - 2.90i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.74 + 6.51i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.71043375775614360154362775090, −10.40832180765074644879584097506, −9.329820409602654887242788566674, −8.546721468514326138122074641235, −7.07766182851503199220961330608, −6.48590083763673834803742365012, −5.29480496686879178756775246391, −4.15820284862048266415613048843, −2.98584036592156409334150289960, −1.68842288472328239694832445460, 0.54720945291227735191618275980, 2.32110256440098384126148376935, 4.41595734933881832853796857471, 5.02979860710654327201553520473, 6.12065353865162653116376901822, 6.76462932701206718801083160562, 7.87266125587032090200655150965, 9.034456817867576347233553705430, 9.282580424145897557792703282818, 10.59285033055588968592734676251

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