Properties

Label 2-570-285.29-c1-0-1
Degree $2$
Conductor $570$
Sign $-0.744 - 0.667i$
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  + (0.984 − 0.173i)2-s + (−1.28 − 1.15i)3-s + (0.939 − 0.342i)4-s + (−0.501 + 2.17i)5-s + (−1.46 − 0.917i)6-s + (−3.34 − 1.93i)7-s + (0.866 − 0.5i)8-s + (0.313 + 2.98i)9-s + (−0.115 + 2.23i)10-s + (−3.47 + 2.00i)11-s + (−1.60 − 0.648i)12-s + (−0.681 − 0.571i)13-s + (−3.63 − 1.32i)14-s + (3.17 − 2.22i)15-s + (0.766 − 0.642i)16-s + (0.566 + 3.21i)17-s + ⋯
L(s)  = 1  + (0.696 − 0.122i)2-s + (−0.743 − 0.669i)3-s + (0.469 − 0.171i)4-s + (−0.224 + 0.974i)5-s + (−0.599 − 0.374i)6-s + (−1.26 − 0.730i)7-s + (0.306 − 0.176i)8-s + (0.104 + 0.994i)9-s + (−0.0366 + 0.706i)10-s + (−1.04 + 0.605i)11-s + (−0.463 − 0.187i)12-s + (−0.189 − 0.158i)13-s + (−0.970 − 0.353i)14-s + (0.818 − 0.573i)15-s + (0.191 − 0.160i)16-s + (0.137 + 0.778i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.744 - 0.667i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.744 - 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.744 - 0.667i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ -0.744 - 0.667i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0859134 + 0.224530i\)
\(L(\frac12)\) \(\approx\) \(0.0859134 + 0.224530i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.984 + 0.173i)T \)
3 \( 1 + (1.28 + 1.15i)T \)
5 \( 1 + (0.501 - 2.17i)T \)
19 \( 1 + (3.42 + 2.69i)T \)
good7 \( 1 + (3.34 + 1.93i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (3.47 - 2.00i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.681 + 0.571i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (-0.566 - 3.21i)T + (-15.9 + 5.81i)T^{2} \)
23 \( 1 + (6.82 - 2.48i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (1.08 - 6.16i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (-1.50 - 0.868i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 2.85T + 37T^{2} \)
41 \( 1 + (-4.43 + 3.72i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-0.819 + 2.25i)T + (-32.9 - 27.6i)T^{2} \)
47 \( 1 + (1.50 - 8.51i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (3.88 + 10.6i)T + (-40.6 + 34.0i)T^{2} \)
59 \( 1 + (-0.00803 - 0.0455i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (2.60 - 0.946i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-0.646 + 3.66i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (11.9 + 4.35i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (7.68 + 9.15i)T + (-12.6 + 71.8i)T^{2} \)
79 \( 1 + (-7.79 - 9.29i)T + (-13.7 + 77.7i)T^{2} \)
83 \( 1 + (1.50 - 2.60i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-6.50 - 5.45i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (-2.39 - 13.5i)T + (-91.1 + 33.1i)T^{2} \)
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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

−10.94081975112933183683620707751, −10.52185518259756062423039530529, −9.813679902066591574333428394279, −7.918272932400568312178048074346, −7.23100612368177726303254639097, −6.49853061865242968037397468206, −5.80027870880477133391408789641, −4.47748627060102399418570733461, −3.32553032931719133971391092963, −2.17811518673173444730890938364, 0.10826563148057781901515355285, 2.66387565388139998696511127852, 3.90920751969188037813908541007, 4.78917033977418226881498620159, 5.89660446082348175336851724280, 6.11792766110446574739934795964, 7.69498518627961810421710239839, 8.734423827340698499681286685255, 9.687920213140740423786360317608, 10.34581586168341146552734816092

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