Properties

Label 2-570-19.4-c1-0-6
Degree $2$
Conductor $570$
Sign $0.204 - 0.978i$
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  + (0.939 + 0.342i)2-s + (0.173 + 0.984i)3-s + (0.766 + 0.642i)4-s + (−0.766 + 0.642i)5-s + (−0.173 + 0.984i)6-s + (1.43 − 2.49i)7-s + (0.500 + 0.866i)8-s + (−0.939 + 0.342i)9-s + (−0.939 + 0.342i)10-s + (2.14 + 3.71i)11-s + (−0.5 + 0.866i)12-s + (−0.794 + 4.50i)13-s + (2.20 − 1.85i)14-s + (−0.766 − 0.642i)15-s + (0.173 + 0.984i)16-s + (5.08 + 1.85i)17-s + ⋯
L(s)  = 1  + (0.664 + 0.241i)2-s + (0.100 + 0.568i)3-s + (0.383 + 0.321i)4-s + (−0.342 + 0.287i)5-s + (−0.0708 + 0.402i)6-s + (0.544 − 0.942i)7-s + (0.176 + 0.306i)8-s + (−0.313 + 0.114i)9-s + (−0.297 + 0.108i)10-s + (0.646 + 1.12i)11-s + (−0.144 + 0.249i)12-s + (−0.220 + 1.24i)13-s + (0.589 − 0.494i)14-s + (−0.197 − 0.165i)15-s + (0.0434 + 0.246i)16-s + (1.23 + 0.448i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.204 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.204 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.204 - 0.978i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.204 - 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.71947 + 1.39710i\)
\(L(\frac12)\) \(\approx\) \(1.71947 + 1.39710i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.939 - 0.342i)T \)
3 \( 1 + (-0.173 - 0.984i)T \)
5 \( 1 + (0.766 - 0.642i)T \)
19 \( 1 + (3.79 + 2.15i)T \)
good7 \( 1 + (-1.43 + 2.49i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.14 - 3.71i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.794 - 4.50i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (-5.08 - 1.85i)T + (13.0 + 10.9i)T^{2} \)
23 \( 1 + (0.0432 + 0.0362i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (-7.59 + 2.76i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (1.88 - 3.26i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 11.5T + 37T^{2} \)
41 \( 1 + (1.37 + 7.80i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (-3.28 + 2.75i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (1.59 - 0.582i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + (3.52 + 2.95i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (5.13 + 1.87i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-7.67 - 6.44i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (-13.1 + 4.79i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (-8.98 + 7.53i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (2.79 + 15.8i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (-0.0778 - 0.441i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-7.82 + 13.5i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.337 - 1.91i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (11.1 + 4.04i)T + (74.3 + 62.3i)T^{2} \)
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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

−10.83844216867321587129120622484, −10.28670152921031433773586024292, −9.198842958256127341334350254663, −8.132975361545871144066113216202, −7.14213731390752689832402766348, −6.57232627430099616805402427736, −5.02939841881568695528218063571, −4.30537436019756683834860160925, −3.59538493517752789035169621668, −1.91264588952327387506524995440, 1.14787200568262356431535023228, 2.68866254669931057061051355636, 3.64157685142095660183382236293, 5.15264377376796712554147751317, 5.73192152821915576536670148661, 6.77994881453705771069196024270, 8.158703339282940934680647027273, 8.401269030255739953067556249654, 9.732107841016175113277715066164, 10.84946330010998652130643274159

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