Properties

Label 2-570-285.284-c1-0-5
Degree $2$
Conductor $570$
Sign $-0.998 - 0.0622i$
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  + i·2-s + (1.41 + i)3-s − 4-s + (−2.12 + 0.707i)5-s + (−1 + 1.41i)6-s i·8-s + (1.00 + 2.82i)9-s + (−0.707 − 2.12i)10-s + 2.82i·11-s + (−1.41 − i)12-s − 4.24·13-s + (−3.70 − 1.12i)15-s + 16-s + (−2.82 + 1.00i)18-s + (1 + 4.24i)19-s + (2.12 − 0.707i)20-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.816 + 0.577i)3-s − 0.5·4-s + (−0.948 + 0.316i)5-s + (−0.408 + 0.577i)6-s − 0.353i·8-s + (0.333 + 0.942i)9-s + (−0.223 − 0.670i)10-s + 0.852i·11-s + (−0.408 − 0.288i)12-s − 1.17·13-s + (−0.957 − 0.289i)15-s + 0.250·16-s + (−0.666 + 0.235i)18-s + (0.229 + 0.973i)19-s + (0.474 − 0.158i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0622i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0622i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.998 - 0.0622i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (569, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ -0.998 - 0.0622i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0351910 + 1.13023i\)
\(L(\frac12)\) \(\approx\) \(0.0351910 + 1.13023i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (-1.41 - i)T \)
5 \( 1 + (2.12 - 0.707i)T \)
19 \( 1 + (-1 - 4.24i)T \)
good7 \( 1 - 7T^{2} \)
11 \( 1 - 2.82iT - 11T^{2} \)
13 \( 1 + 4.24T + 13T^{2} \)
17 \( 1 + 17T^{2} \)
23 \( 1 + 4.24T + 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + 4.24iT - 31T^{2} \)
37 \( 1 + 4.24T + 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - 12iT - 43T^{2} \)
47 \( 1 - 12.7T + 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 + 6T + 59T^{2} \)
61 \( 1 - 10T + 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 - 6iT - 73T^{2} \)
79 \( 1 - 12.7iT - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 12T + 89T^{2} \)
97 \( 1 + 97T^{2} \)
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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

−11.00658377494151154910804321557, −9.954624702111888940207214955766, −9.478973812713731920560581582203, −8.273043378636069290392746872663, −7.65220600239859273594896480910, −7.06533608656079239618441658940, −5.55709104935704091131346245771, −4.42887746046340257859912100042, −3.78894507740846766411173293716, −2.38062397866808396091169083585, 0.57445133394908193928306761427, 2.26944524551373368276288639410, 3.35421839489797830890578378539, 4.24980664160823953742159140562, 5.52106008817624816236439520742, 7.09320944726372468239357634226, 7.68889498634096737755418216616, 8.734138698715187983563810281700, 9.176070961563806686164754602401, 10.36700221175778596063267785054

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