Properties

Label 2-570-95.37-c1-0-19
Degree $2$
Conductor $570$
Sign $-0.995 - 0.0938i$
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 + (−0.707 − 0.707i)3-s − 1.00i·4-s + (0.114 − 2.23i)5-s − 1.00·6-s + (−1.40 − 1.40i)7-s + (−0.707 − 0.707i)8-s + 1.00i·9-s + (−1.49 − 1.66i)10-s − 5.29·11-s + (−0.707 + 0.707i)12-s + (1.91 + 1.91i)13-s − 1.98·14-s + (−1.66 + 1.49i)15-s − 1.00·16-s + (0.488 + 0.488i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−0.408 − 0.408i)3-s − 0.500i·4-s + (0.0512 − 0.998i)5-s − 0.408·6-s + (−0.530 − 0.530i)7-s + (−0.250 − 0.250i)8-s + 0.333i·9-s + (−0.473 − 0.524i)10-s − 1.59·11-s + (−0.204 + 0.204i)12-s + (0.532 + 0.532i)13-s − 0.530·14-s + (−0.428 + 0.386i)15-s − 0.250·16-s + (0.118 + 0.118i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0938i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.995 - 0.0938i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ -0.995 - 0.0938i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0489429 + 1.04052i\)
\(L(\frac12)\) \(\approx\) \(0.0489429 + 1.04052i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 + (-0.114 + 2.23i)T \)
19 \( 1 + (-2.44 + 3.60i)T \)
good7 \( 1 + (1.40 + 1.40i)T + 7iT^{2} \)
11 \( 1 + 5.29T + 11T^{2} \)
13 \( 1 + (-1.91 - 1.91i)T + 13iT^{2} \)
17 \( 1 + (-0.488 - 0.488i)T + 17iT^{2} \)
23 \( 1 + (3.88 - 3.88i)T - 23iT^{2} \)
29 \( 1 - 6.19T + 29T^{2} \)
31 \( 1 + 7.84iT - 31T^{2} \)
37 \( 1 + (5.60 - 5.60i)T - 37iT^{2} \)
41 \( 1 - 1.90iT - 41T^{2} \)
43 \( 1 + (-7.12 + 7.12i)T - 43iT^{2} \)
47 \( 1 + (7.86 + 7.86i)T + 47iT^{2} \)
53 \( 1 + (8.93 + 8.93i)T + 53iT^{2} \)
59 \( 1 - 0.611T + 59T^{2} \)
61 \( 1 - 8.31T + 61T^{2} \)
67 \( 1 + (10.2 - 10.2i)T - 67iT^{2} \)
71 \( 1 + 3.97iT - 71T^{2} \)
73 \( 1 + (-7.72 + 7.72i)T - 73iT^{2} \)
79 \( 1 - 17.1T + 79T^{2} \)
83 \( 1 + (-5.43 + 5.43i)T - 83iT^{2} \)
89 \( 1 + 5.42T + 89T^{2} \)
97 \( 1 + (-3.10 + 3.10i)T - 97iT^{2} \)
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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

−10.30914386237267187594204559294, −9.698778889726366880020631044464, −8.493541119909932679156184500942, −7.59809726577563129706138613796, −6.45867892272519556230366172622, −5.45331162672999607767527693087, −4.73754956285813435817675748993, −3.51098706262514441943827181711, −2.03997196886770508374899454821, −0.50595066137470423855929677772, 2.68304439374737822393780073039, 3.43624368765826543489357890110, 4.86228843204681704109481467117, 5.81929487402356892516283233106, 6.37303289123228612838638341887, 7.55991861568824210509817472082, 8.299832607381394910326413330539, 9.620002616968968531473299443982, 10.48753828786690296808483558784, 10.95688323781262920252569712261

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