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Properties

Label 2-570-15.8-c1-0-33
Degree $2$
Conductor $570$
Sign $-0.998 - 0.0596i$
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 + (−0.752 − 1.55i)3-s + 1.00i·4-s + (1.50 − 1.64i)5-s + (−0.570 + 1.63i)6-s + (0.306 − 0.306i)7-s + (0.707 − 0.707i)8-s + (−1.86 + 2.34i)9-s + (−2.23 + 0.0989i)10-s − 0.944i·11-s + (1.55 − 0.752i)12-s + (−4.86 − 4.86i)13-s − 0.433·14-s + (−3.70 − 1.11i)15-s − 1.00·16-s + (2.18 + 2.18i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−0.434 − 0.900i)3-s + 0.500i·4-s + (0.675 − 0.737i)5-s + (−0.232 + 0.667i)6-s + (0.115 − 0.115i)7-s + (0.250 − 0.250i)8-s + (−0.622 + 0.782i)9-s + (−0.706 + 0.0312i)10-s − 0.284i·11-s + (0.450 − 0.217i)12-s + (−1.34 − 1.34i)13-s − 0.115·14-s + (−0.957 − 0.287i)15-s − 0.250·16-s + (0.529 + 0.529i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0596i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0596i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.998 - 0.0596i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (533, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ -0.998 - 0.0596i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0231512 + 0.776117i\)
\(L(\frac12)\) \(\approx\) \(0.0231512 + 0.776117i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.752 + 1.55i)T \)
5 \( 1 + (-1.50 + 1.64i)T \)
19 \( 1 + iT \)
good7 \( 1 + (-0.306 + 0.306i)T - 7iT^{2} \)
11 \( 1 + 0.944iT - 11T^{2} \)
13 \( 1 + (4.86 + 4.86i)T + 13iT^{2} \)
17 \( 1 + (-2.18 - 2.18i)T + 17iT^{2} \)
23 \( 1 + (-5.00 + 5.00i)T - 23iT^{2} \)
29 \( 1 + 1.77T + 29T^{2} \)
31 \( 1 + 10.9T + 31T^{2} \)
37 \( 1 + (3.09 - 3.09i)T - 37iT^{2} \)
41 \( 1 - 6.08iT - 41T^{2} \)
43 \( 1 + (3.19 + 3.19i)T + 43iT^{2} \)
47 \( 1 + (4.22 + 4.22i)T + 47iT^{2} \)
53 \( 1 + (5.08 - 5.08i)T - 53iT^{2} \)
59 \( 1 - 7.52T + 59T^{2} \)
61 \( 1 - 4.53T + 61T^{2} \)
67 \( 1 + (-7.65 + 7.65i)T - 67iT^{2} \)
71 \( 1 + 12.9iT - 71T^{2} \)
73 \( 1 + (-2.53 - 2.53i)T + 73iT^{2} \)
79 \( 1 - 9.50iT - 79T^{2} \)
83 \( 1 + (-7.84 + 7.84i)T - 83iT^{2} \)
89 \( 1 + 8.88T + 89T^{2} \)
97 \( 1 + (0.722 - 0.722i)T - 97iT^{2} \)
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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

−10.38898684619011465367930306414, −9.469343524498232489288955947117, −8.460376249503796675837467352657, −7.77405396777839582095953722195, −6.80022703881198876990714435580, −5.58533361918866045170960959302, −4.90784992020687008577576082047, −3.01691407707628949040104331338, −1.83732170231650393483829972417, −0.53281721754875064570775300823, 2.05699557780660591589495956628, 3.57326817667806179139534028852, 5.01903002349199810294232809229, 5.57189694249808200072409415349, 6.85469235638189474062154275876, 7.30755717200480936998590741702, 8.903510961576901046614770045673, 9.607015523106737519844319975632, 9.964822237481043839755088880428, 11.11034452024552027988409427309

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