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Properties

Label 2-570-95.8-c1-0-11
Degree $2$
Conductor $570$
Sign $0.994 - 0.100i$
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 + (2.10 − 0.754i)5-s + (0.499 + 0.866i)6-s + (−0.506 + 0.506i)7-s + (0.707 − 0.707i)8-s + (−0.866 + 0.499i)9-s + (1.83 − 1.27i)10-s − 1.16·11-s + (0.707 + 0.707i)12-s + (6.66 + 1.78i)13-s + (−0.358 + 0.620i)14-s + (1.27 + 1.83i)15-s + (0.500 − 0.866i)16-s + (−0.465 − 1.73i)17-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (0.149 + 0.557i)3-s + (0.433 − 0.249i)4-s + (0.941 − 0.337i)5-s + (0.204 + 0.353i)6-s + (−0.191 + 0.191i)7-s + (0.249 − 0.249i)8-s + (−0.288 + 0.166i)9-s + (0.581 − 0.402i)10-s − 0.350·11-s + (0.204 + 0.204i)12-s + (1.84 + 0.495i)13-s + (−0.0957 + 0.165i)14-s + (0.328 + 0.474i)15-s + (0.125 − 0.216i)16-s + (−0.112 − 0.421i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.100i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 - 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.994 - 0.100i$
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: \((2,\ 570,\ (\ :1/2),\ 0.994 - 0.100i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.62666 + 0.132385i\)
\(L(\frac12)\) \(\approx\) \(2.62666 + 0.132385i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.10 + 0.754i)T \)
19 \( 1 + (4.14 + 1.33i)T \)
good7 \( 1 + (0.506 - 0.506i)T - 7iT^{2} \)
11 \( 1 + 1.16T + 11T^{2} \)
13 \( 1 + (-6.66 - 1.78i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (0.465 + 1.73i)T + (-14.7 + 8.5i)T^{2} \)
23 \( 1 + (0.517 - 1.93i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-2.32 - 4.03i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 0.506iT - 31T^{2} \)
37 \( 1 + (3.08 + 3.08i)T + 37iT^{2} \)
41 \( 1 + (2.46 + 1.42i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.03 - 1.08i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (1.65 + 0.443i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (9.62 + 2.57i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-6.04 + 10.4i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.75 - 13.4i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.91 + 7.13i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (6.60 + 3.81i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (2.63 - 0.707i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (7.23 - 12.5i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (9.66 + 9.66i)T + 83iT^{2} \)
89 \( 1 + (2.91 + 5.05i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.21 + 0.861i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.79793693940023424412276968588, −9.989844066920121709277519245783, −9.027771279524873227539135222169, −8.413041611170052053453592020607, −6.77583054797315805751474063030, −6.00553770704914323763633802795, −5.14800576641808054256439657633, −4.12446627020314056181352568837, −2.99684381844538693093074742152, −1.68464561363577982397826916018, 1.60367741052036589316876212589, 2.86231504609087759523824516157, 3.94641398226489097542368265178, 5.41201520807975726254183725771, 6.27202709025432201312125015550, 6.70750109133933492503729822456, 8.096063430982226633603841289348, 8.689506392950283065457794766919, 10.10864794242506559952280743492, 10.71539951644807292935993876277

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