Properties

Label 2-570-15.2-c1-0-7
Degree $2$
Conductor $570$
Sign $-0.0731 - 0.997i$
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 + (1.43 + 0.966i)3-s − 1.00i·4-s + (−1.74 − 1.39i)5-s + (−1.69 + 0.333i)6-s + (0.371 + 0.371i)7-s + (0.707 + 0.707i)8-s + (1.13 + 2.77i)9-s + (2.22 − 0.248i)10-s + 4.73i·11-s + (0.966 − 1.43i)12-s + (3.07 − 3.07i)13-s − 0.524·14-s + (−1.16 − 3.69i)15-s − 1.00·16-s + (−2.77 + 2.77i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (0.829 + 0.557i)3-s − 0.500i·4-s + (−0.781 − 0.624i)5-s + (−0.693 + 0.136i)6-s + (0.140 + 0.140i)7-s + (0.250 + 0.250i)8-s + (0.377 + 0.925i)9-s + (0.702 − 0.0785i)10-s + 1.42i·11-s + (0.278 − 0.414i)12-s + (0.852 − 0.852i)13-s − 0.140·14-s + (−0.300 − 0.953i)15-s − 0.250·16-s + (−0.672 + 0.672i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0731 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0731 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.0731 - 0.997i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ -0.0731 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.886566 + 0.953961i\)
\(L(\frac12)\) \(\approx\) \(0.886566 + 0.953961i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.43 - 0.966i)T \)
5 \( 1 + (1.74 + 1.39i)T \)
19 \( 1 - iT \)
good7 \( 1 + (-0.371 - 0.371i)T + 7iT^{2} \)
11 \( 1 - 4.73iT - 11T^{2} \)
13 \( 1 + (-3.07 + 3.07i)T - 13iT^{2} \)
17 \( 1 + (2.77 - 2.77i)T - 17iT^{2} \)
23 \( 1 + (-5.62 - 5.62i)T + 23iT^{2} \)
29 \( 1 - 4.06T + 29T^{2} \)
31 \( 1 - 3.53T + 31T^{2} \)
37 \( 1 + (-2.20 - 2.20i)T + 37iT^{2} \)
41 \( 1 + 6.41iT - 41T^{2} \)
43 \( 1 + (6.80 - 6.80i)T - 43iT^{2} \)
47 \( 1 + (5.86 - 5.86i)T - 47iT^{2} \)
53 \( 1 + (6.65 + 6.65i)T + 53iT^{2} \)
59 \( 1 - 6.00T + 59T^{2} \)
61 \( 1 + 3.00T + 61T^{2} \)
67 \( 1 + (-4.01 - 4.01i)T + 67iT^{2} \)
71 \( 1 + 3.67iT - 71T^{2} \)
73 \( 1 + (-7.51 + 7.51i)T - 73iT^{2} \)
79 \( 1 + 7.93iT - 79T^{2} \)
83 \( 1 + (7.74 + 7.74i)T + 83iT^{2} \)
89 \( 1 - 7.13T + 89T^{2} \)
97 \( 1 + (-5.01 - 5.01i)T + 97iT^{2} \)
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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

−10.72670434558361799658811135892, −9.866287581389880305746775456408, −9.078887740679253347027575643552, −8.266845880557848840856422087780, −7.78092785146254983811669376054, −6.71835215783471964462309504854, −5.19635497965501446115589401370, −4.47391162055656048729364533541, −3.31292880302429385094108340056, −1.61213816136139132683437054838, 0.875658315803429332000946992547, 2.59993654907193524657562498764, 3.37670881674209969781743584336, 4.43119628811036897815910425535, 6.48427757332766572794074558997, 6.94523370900797341677468455093, 8.243761123595469187097728096725, 8.514683918956608059758713169855, 9.447933836199028972240931989503, 10.74067050878610330158606931048

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