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Properties

Label 2-570-95.8-c1-0-14
Degree $2$
Conductor $570$
Sign $-0.873 + 0.486i$
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  + (−0.965 + 0.258i)2-s + (−0.258 − 0.965i)3-s + (0.866 − 0.499i)4-s + (0.392 + 2.20i)5-s + (0.499 + 0.866i)6-s + (−3.01 + 3.01i)7-s + (−0.707 + 0.707i)8-s + (−0.866 + 0.499i)9-s + (−0.948 − 2.02i)10-s + 0.889·11-s + (−0.707 − 0.707i)12-s + (−3.48 − 0.935i)13-s + (2.12 − 3.68i)14-s + (2.02 − 0.948i)15-s + (0.500 − 0.866i)16-s + (−2.12 − 7.93i)17-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (−0.149 − 0.557i)3-s + (0.433 − 0.249i)4-s + (0.175 + 0.984i)5-s + (0.204 + 0.353i)6-s + (−1.13 + 1.13i)7-s + (−0.249 + 0.249i)8-s + (−0.288 + 0.166i)9-s + (−0.299 − 0.640i)10-s + 0.268·11-s + (−0.204 − 0.204i)12-s + (−0.967 − 0.259i)13-s + (0.569 − 0.985i)14-s + (0.522 − 0.244i)15-s + (0.125 − 0.216i)16-s + (−0.515 − 1.92i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.873 + 0.486i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.873 + 0.486i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.873 + 0.486i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ -0.873 + 0.486i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0174801 - 0.0673066i\)
\(L(\frac12)\) \(\approx\) \(0.0174801 - 0.0673066i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 - 0.258i)T \)
3 \( 1 + (0.258 + 0.965i)T \)
5 \( 1 + (-0.392 - 2.20i)T \)
19 \( 1 + (1.13 + 4.20i)T \)
good7 \( 1 + (3.01 - 3.01i)T - 7iT^{2} \)
11 \( 1 - 0.889T + 11T^{2} \)
13 \( 1 + (3.48 + 0.935i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (2.12 + 7.93i)T + (-14.7 + 8.5i)T^{2} \)
23 \( 1 + (-0.772 + 2.88i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (3.84 + 6.66i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.64iT - 31T^{2} \)
37 \( 1 + (-3.39 - 3.39i)T + 37iT^{2} \)
41 \( 1 + (-4.93 - 2.84i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (11.1 - 2.98i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (9.22 + 2.47i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-10.6 - 2.86i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (1.41 - 2.45i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.57 + 4.46i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.57 - 5.87i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (1.67 + 0.965i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (3.11 - 0.835i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (5.06 - 8.77i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.14 + 4.14i)T + 83iT^{2} \)
89 \( 1 + (4.50 + 7.80i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.59 + 0.426i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.08712393326027373518824899878, −9.541902579946373152724977532881, −8.745943902590476086411644723485, −7.46647051030370974992235712168, −6.78565243854801312547442560222, −6.20010345033298446170488073827, −5.04878264981230202454982022883, −2.88037001715341623329970244776, −2.46570307989251521312232035906, −0.04701410789876875616374893194, 1.68510758953209204416622625517, 3.58579162421346570683642345673, 4.26572841382502646316425969239, 5.69698688319846534058868895288, 6.64329118891295265009941982396, 7.69254267158039632202355836494, 8.694925682459896192967076248568, 9.523551661646375512413247785666, 10.08770684422813896711113413275, 10.76481804928314478202347147826

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