Properties

Label 2-570-285.47-c1-0-20
Degree $2$
Conductor $570$
Sign $0.391 - 0.920i$
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  + (0.819 + 0.573i)2-s + (1.71 − 0.236i)3-s + (0.342 + 0.939i)4-s + (0.250 + 2.22i)5-s + (1.54 + 0.790i)6-s + (0.373 − 0.100i)7-s + (−0.258 + 0.965i)8-s + (2.88 − 0.810i)9-s + (−1.06 + 1.96i)10-s + (−3.61 + 2.08i)11-s + (0.808 + 1.53i)12-s + (0.240 − 0.0210i)13-s + (0.363 + 0.132i)14-s + (0.954 + 3.75i)15-s + (−0.766 + 0.642i)16-s + (−0.0718 − 0.0503i)17-s + ⋯
L(s)  = 1  + (0.579 + 0.405i)2-s + (0.990 − 0.136i)3-s + (0.171 + 0.469i)4-s + (0.112 + 0.993i)5-s + (0.629 + 0.322i)6-s + (0.141 − 0.0378i)7-s + (−0.0915 + 0.341i)8-s + (0.962 − 0.270i)9-s + (−0.338 + 0.621i)10-s + (−1.09 + 0.629i)11-s + (0.233 + 0.442i)12-s + (0.0667 − 0.00583i)13-s + (0.0971 + 0.0353i)14-s + (0.246 + 0.969i)15-s + (−0.191 + 0.160i)16-s + (−0.0174 − 0.0122i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.391 - 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.391 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.391 - 0.920i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.391 - 0.920i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.29118 + 1.51544i\)
\(L(\frac12)\) \(\approx\) \(2.29118 + 1.51544i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.819 - 0.573i)T \)
3 \( 1 + (-1.71 + 0.236i)T \)
5 \( 1 + (-0.250 - 2.22i)T \)
19 \( 1 + (-3.07 + 3.08i)T \)
good7 \( 1 + (-0.373 + 0.100i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (3.61 - 2.08i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.240 + 0.0210i)T + (12.8 - 2.25i)T^{2} \)
17 \( 1 + (0.0718 + 0.0503i)T + (5.81 + 15.9i)T^{2} \)
23 \( 1 + (-1.28 - 0.598i)T + (14.7 + 17.6i)T^{2} \)
29 \( 1 + (-0.577 + 3.27i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (-2.14 + 3.71i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (7.64 + 7.64i)T + 37iT^{2} \)
41 \( 1 + (-3.48 - 4.15i)T + (-7.11 + 40.3i)T^{2} \)
43 \( 1 + (-2.61 + 1.22i)T + (27.6 - 32.9i)T^{2} \)
47 \( 1 + (0.0805 + 0.115i)T + (-16.0 + 44.1i)T^{2} \)
53 \( 1 + (-6.98 - 3.25i)T + (34.0 + 40.6i)T^{2} \)
59 \( 1 + (0.127 + 0.724i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-6.62 + 2.40i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-0.994 + 0.696i)T + (22.9 - 62.9i)T^{2} \)
71 \( 1 + (-1.77 + 4.86i)T + (-54.3 - 45.6i)T^{2} \)
73 \( 1 + (-0.355 + 4.05i)T + (-71.8 - 12.6i)T^{2} \)
79 \( 1 + (9.94 + 11.8i)T + (-13.7 + 77.7i)T^{2} \)
83 \( 1 + (8.49 - 2.27i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (-1.19 - 1.00i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (6.33 - 9.04i)T + (-33.1 - 91.1i)T^{2} \)
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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

−10.83547752893196265865890232635, −10.00304779803493235677941790497, −9.100424086465592287729140914163, −7.85500423498623891858874881256, −7.43351709029480879810373827695, −6.56036768909542798432378898637, −5.32755855563407182529320826920, −4.16231933617084567911748389298, −3.03158600879697914643147961759, −2.24422685387232336810465244517, 1.38343378532801051672696728948, 2.73511840452037836533558036330, 3.75213085774720333500723004928, 4.89700054166583617248725899415, 5.57495135247799982371779803765, 7.07487317205415122932069786098, 8.222529893798289379324635265906, 8.684739546268924234076519336117, 9.817693982851412776389938573113, 10.41294557022657083032513862320

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