Properties

Label 2-570-15.8-c1-0-1
Degree $2$
Conductor $570$
Sign $-0.396 + 0.918i$
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  + (0.707 + 0.707i)2-s + (−0.518 + 1.65i)3-s + 1.00i·4-s + (−2.11 − 0.736i)5-s + (−1.53 + 0.801i)6-s + (−2.44 + 2.44i)7-s + (−0.707 + 0.707i)8-s + (−2.46 − 1.71i)9-s + (−0.972 − 2.01i)10-s − 5.62i·11-s + (−1.65 − 0.518i)12-s + (0.933 + 0.933i)13-s − 3.45·14-s + (2.31 − 3.10i)15-s − 1.00·16-s + (0.0532 + 0.0532i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (−0.299 + 0.954i)3-s + 0.500i·4-s + (−0.944 − 0.329i)5-s + (−0.626 + 0.327i)6-s + (−0.922 + 0.922i)7-s + (−0.250 + 0.250i)8-s + (−0.820 − 0.571i)9-s + (−0.307 − 0.636i)10-s − 1.69i·11-s + (−0.477 − 0.149i)12-s + (0.259 + 0.259i)13-s − 0.922·14-s + (0.596 − 0.802i)15-s − 0.250·16-s + (0.0129 + 0.0129i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.396 + 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.396 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.396 + 0.918i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (533, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ -0.396 + 0.918i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.110549 - 0.168164i\)
\(L(\frac12)\) \(\approx\) \(0.110549 - 0.168164i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
3 \( 1 + (0.518 - 1.65i)T \)
5 \( 1 + (2.11 + 0.736i)T \)
19 \( 1 + iT \)
good7 \( 1 + (2.44 - 2.44i)T - 7iT^{2} \)
11 \( 1 + 5.62iT - 11T^{2} \)
13 \( 1 + (-0.933 - 0.933i)T + 13iT^{2} \)
17 \( 1 + (-0.0532 - 0.0532i)T + 17iT^{2} \)
23 \( 1 + (0.841 - 0.841i)T - 23iT^{2} \)
29 \( 1 + 6.56T + 29T^{2} \)
31 \( 1 + 5.11T + 31T^{2} \)
37 \( 1 + (5.88 - 5.88i)T - 37iT^{2} \)
41 \( 1 - 5.75iT - 41T^{2} \)
43 \( 1 + (7.57 + 7.57i)T + 43iT^{2} \)
47 \( 1 + (-2.64 - 2.64i)T + 47iT^{2} \)
53 \( 1 + (9.67 - 9.67i)T - 53iT^{2} \)
59 \( 1 + 11.0T + 59T^{2} \)
61 \( 1 - 9.82T + 61T^{2} \)
67 \( 1 + (-6.27 + 6.27i)T - 67iT^{2} \)
71 \( 1 + 2.29iT - 71T^{2} \)
73 \( 1 + (7.35 + 7.35i)T + 73iT^{2} \)
79 \( 1 + 3.41iT - 79T^{2} \)
83 \( 1 + (-4.84 + 4.84i)T - 83iT^{2} \)
89 \( 1 - 4.69T + 89T^{2} \)
97 \( 1 + (4.18 - 4.18i)T - 97iT^{2} \)
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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

−11.45041778062252053630463635414, −10.66709295700003694252902649918, −9.216140120439140337492948879474, −8.866656477821851834187968233230, −7.907536969016331898724517945985, −6.48810544506822005162999939559, −5.78461845909586848227435210737, −4.92995170122649643828985205341, −3.62424147405396598077111937678, −3.20201058358360508648791181855, 0.096362737108088116924926828919, 1.87681484582221759990642830034, 3.31789042641333024668607757738, 4.20642088560751415592474497652, 5.45847581799265597612655084128, 6.84430067004457056477515306913, 7.08633944452114724242430761474, 8.076951092094181022851856715756, 9.502514229648751959914792817513, 10.41273875995143942018995841357

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