Properties

Label 2-570-15.2-c1-0-35
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: $\chi_{570} (77, \cdot )$
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

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

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