Properties

Label 2-570-285.29-c1-0-36
Degree $2$
Conductor $570$
Sign $0.631 + 0.775i$
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  + (0.984 − 0.173i)2-s + (1.20 − 1.24i)3-s + (0.939 − 0.342i)4-s + (2.18 − 0.487i)5-s + (0.974 − 1.43i)6-s + (1.28 + 0.739i)7-s + (0.866 − 0.5i)8-s + (−0.0805 − 2.99i)9-s + (2.06 − 0.858i)10-s + (−2.81 + 1.62i)11-s + (0.710 − 1.57i)12-s + (−3.29 − 2.76i)13-s + (1.38 + 0.505i)14-s + (2.03 − 3.29i)15-s + (0.766 − 0.642i)16-s + (0.748 + 4.24i)17-s + ⋯
L(s)  = 1  + (0.696 − 0.122i)2-s + (0.697 − 0.716i)3-s + (0.469 − 0.171i)4-s + (0.975 − 0.217i)5-s + (0.397 − 0.584i)6-s + (0.483 + 0.279i)7-s + (0.306 − 0.176i)8-s + (−0.0268 − 0.999i)9-s + (0.652 − 0.271i)10-s + (−0.849 + 0.490i)11-s + (0.205 − 0.455i)12-s + (−0.913 − 0.766i)13-s + (0.371 + 0.135i)14-s + (0.524 − 0.851i)15-s + (0.191 − 0.160i)16-s + (0.181 + 1.02i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.631 + 0.775i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.631 + 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.631 + 0.775i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.631 + 0.775i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.76908 - 1.31641i\)
\(L(\frac12)\) \(\approx\) \(2.76908 - 1.31641i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.984 + 0.173i)T \)
3 \( 1 + (-1.20 + 1.24i)T \)
5 \( 1 + (-2.18 + 0.487i)T \)
19 \( 1 + (-1.45 - 4.10i)T \)
good7 \( 1 + (-1.28 - 0.739i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.81 - 1.62i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.29 + 2.76i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (-0.748 - 4.24i)T + (-15.9 + 5.81i)T^{2} \)
23 \( 1 + (7.23 - 2.63i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (0.791 - 4.48i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (-2.19 - 1.26i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 4.11T + 37T^{2} \)
41 \( 1 + (-1.02 + 0.862i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-2.81 + 7.73i)T + (-32.9 - 27.6i)T^{2} \)
47 \( 1 + (-2.06 + 11.6i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (0.503 + 1.38i)T + (-40.6 + 34.0i)T^{2} \)
59 \( 1 + (-2.10 - 11.9i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-8.53 + 3.10i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (2.31 - 13.1i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-2.07 - 0.757i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (7.62 + 9.08i)T + (-12.6 + 71.8i)T^{2} \)
79 \( 1 + (-4.55 - 5.43i)T + (-13.7 + 77.7i)T^{2} \)
83 \( 1 + (1.29 - 2.23i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (3.44 + 2.89i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (-1.63 - 9.26i)T + (-91.1 + 33.1i)T^{2} \)
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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

−10.28381396973243364066839199910, −10.06340231733054540272312952112, −8.682175438910003520610231751763, −7.902265616451503234677450371776, −7.03370438348567253589377801883, −5.78330128470879967966245618941, −5.27556309753238726691307386634, −3.75290465746985167681752056674, −2.44639951317079163354128832177, −1.71016733396710459916649524073, 2.21918209324773764823956331450, 2.93160635102614624316713970282, 4.45157956813876979671487136525, 5.04019031351365924048269386385, 6.11787044678728180852084820640, 7.33748184351042924463460265288, 8.102640412445505164993053981343, 9.351485645524650970830720241897, 9.899581075581478938513919522588, 10.84456636310399819282370581206

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