Properties

Label 2-570-285.68-c1-0-7
Degree $2$
Conductor $570$
Sign $0.468 - 0.883i$
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  + (0.965 − 0.258i)2-s + (−0.724 − 1.57i)3-s + (0.866 − 0.499i)4-s + (−0.448 + 2.19i)5-s + (−1.10 − 1.33i)6-s + (−2.12 + 2.12i)7-s + (0.707 − 0.707i)8-s + (−1.94 + 2.28i)9-s + (0.133 + 2.23i)10-s + 5.82i·11-s + (−1.41 − 1.00i)12-s + (−0.732 + 2.73i)13-s + (−1.5 + 2.59i)14-s + (3.77 − 0.882i)15-s + (0.500 − 0.866i)16-s + (6.02 − 1.61i)17-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (−0.418 − 0.908i)3-s + (0.433 − 0.249i)4-s + (−0.200 + 0.979i)5-s + (−0.452 − 0.543i)6-s + (−0.801 + 0.801i)7-s + (0.249 − 0.249i)8-s + (−0.649 + 0.760i)9-s + (0.0423 + 0.705i)10-s + 1.75i·11-s + (−0.408 − 0.288i)12-s + (−0.203 + 0.757i)13-s + (−0.400 + 0.694i)14-s + (0.973 − 0.227i)15-s + (0.125 − 0.216i)16-s + (1.46 − 0.391i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.468 - 0.883i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.468 - 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.468 - 0.883i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.468 - 0.883i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16963 + 0.703646i\)
\(L(\frac12)\) \(\approx\) \(1.16963 + 0.703646i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 + 0.258i)T \)
3 \( 1 + (0.724 + 1.57i)T \)
5 \( 1 + (0.448 - 2.19i)T \)
19 \( 1 + (4.33 + 0.5i)T \)
good7 \( 1 + (2.12 - 2.12i)T - 7iT^{2} \)
11 \( 1 - 5.82iT - 11T^{2} \)
13 \( 1 + (0.732 - 2.73i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + (-6.02 + 1.61i)T + (14.7 - 8.5i)T^{2} \)
23 \( 1 + (-2.89 - 0.776i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (4.41 + 7.64i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 2.24T + 31T^{2} \)
37 \( 1 + (-2.12 + 2.12i)T - 37iT^{2} \)
41 \( 1 + (-2.59 - 1.5i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.09 - 1.09i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (2.71 - 10.1i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-7.23 - 1.93i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (2.82 - 4.89i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.12 - 5.40i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.46 - 1.46i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-11.6 - 6.70i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-7.49 + 2.00i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (9.08 + 5.24i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3 + 3i)T - 83iT^{2} \)
89 \( 1 + (3.08 + 5.34i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.643 - 2.40i)T + (-84.0 + 48.5i)T^{2} \)
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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

−11.22162391711010624106921729854, −10.10074034975790110069255447314, −9.404836628839553173918097842562, −7.76106454373368106296573242375, −7.10700041953528192887744633056, −6.37880512070809876201258843799, −5.54795278510544496082387969534, −4.26198958317951763621462733550, −2.81880785083326308845506338415, −2.03067000044693190477455090093, 0.64166356847407858874519237539, 3.44016957811047403638438859677, 3.68966586640603983749563542354, 5.14620790058793070954252806497, 5.64823838886900539415950827589, 6.67510620919694001522920575769, 8.051550322018948619701877153629, 8.775503312756140798816852427053, 9.882650561480301965324836758980, 10.67446807988434557171851343433

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