Properties

Label 2-570-15.2-c1-0-2
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: $\chi_{570} (77, \cdot )$
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

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

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