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Properties

Label 2-570-95.88-c1-0-9
Degree $2$
Conductor $570$
Sign $0.613 - 0.789i$
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  + (−0.258 + 0.965i)2-s + (0.965 + 0.258i)3-s + (−0.866 − 0.499i)4-s + (−1.90 − 1.17i)5-s + (−0.499 + 0.866i)6-s + (−0.515 + 0.515i)7-s + (0.707 − 0.707i)8-s + (0.866 + 0.499i)9-s + (1.62 − 1.53i)10-s + 6.29·11-s + (−0.707 − 0.707i)12-s + (0.216 + 0.806i)13-s + (−0.364 − 0.631i)14-s + (−1.53 − 1.62i)15-s + (0.500 + 0.866i)16-s + (−1.35 − 0.362i)17-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (0.557 + 0.149i)3-s + (−0.433 − 0.249i)4-s + (−0.850 − 0.526i)5-s + (−0.204 + 0.353i)6-s + (−0.194 + 0.194i)7-s + (0.249 − 0.249i)8-s + (0.288 + 0.166i)9-s + (0.515 − 0.484i)10-s + 1.89·11-s + (−0.204 − 0.204i)12-s + (0.0599 + 0.223i)13-s + (−0.0974 − 0.168i)14-s + (−0.395 − 0.420i)15-s + (0.125 + 0.216i)16-s + (−0.327 − 0.0878i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.613 - 0.789i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.613 - 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.613 - 0.789i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.613 - 0.789i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.30726 + 0.640044i\)
\(L(\frac12)\) \(\approx\) \(1.30726 + 0.640044i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 - 0.965i)T \)
3 \( 1 + (-0.965 - 0.258i)T \)
5 \( 1 + (1.90 + 1.17i)T \)
19 \( 1 + (-3.82 - 2.09i)T \)
good7 \( 1 + (0.515 - 0.515i)T - 7iT^{2} \)
11 \( 1 - 6.29T + 11T^{2} \)
13 \( 1 + (-0.216 - 0.806i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (1.35 + 0.362i)T + (14.7 + 8.5i)T^{2} \)
23 \( 1 + (-5.74 + 1.53i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (0.900 - 1.55i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 6.93iT - 31T^{2} \)
37 \( 1 + (-0.856 - 0.856i)T + 37iT^{2} \)
41 \( 1 + (-4.31 + 2.49i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.77 + 6.63i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (1.14 + 4.26i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (3.49 + 13.0i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-0.106 - 0.183i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.48 - 2.57i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (8.34 - 2.23i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-6.29 + 3.63i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (1.64 - 6.15i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (1.71 + 2.97i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.52 - 7.52i)T + 83iT^{2} \)
89 \( 1 + (-3.15 + 5.45i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.05 + 3.93i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.84172916599669795698116770316, −9.487903762873247511915547191615, −9.026959216670039288562275324277, −8.366197439498014561217177767660, −7.24599020980027800168117469198, −6.63276296532368666159097407018, −5.25339527294084392712660808894, −4.21504372178365773311280749903, −3.38490966735256492708349138494, −1.26965275372266799678559865559, 1.11539927181665101552391763194, 2.80205974881211696836375814406, 3.69374513855580377268238443401, 4.48443709976366667425601712865, 6.28584845493512079771148266176, 7.20861019465699893815129170054, 7.972923021996454783043517529021, 9.166968636065161838722188137606, 9.476723708312703493436803028606, 10.81484597769990358272296659512

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