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Properties

Label 2-570-95.27-c1-0-17
Degree $2$
Conductor $570$
Sign $-0.929 + 0.368i$
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  + (−0.258 − 0.965i)2-s + (−0.965 + 0.258i)3-s + (−0.866 + 0.499i)4-s + (1.05 − 1.96i)5-s + (0.499 + 0.866i)6-s + (−2.04 − 2.04i)7-s + (0.707 + 0.707i)8-s + (0.866 − 0.499i)9-s + (−2.17 − 0.513i)10-s + 5.53·11-s + (0.707 − 0.707i)12-s + (0.692 − 2.58i)13-s + (−1.44 + 2.49i)14-s + (−0.513 + 2.17i)15-s + (0.500 − 0.866i)16-s + (−5.09 + 1.36i)17-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s + (−0.557 + 0.149i)3-s + (−0.433 + 0.249i)4-s + (0.473 − 0.880i)5-s + (0.204 + 0.353i)6-s + (−0.771 − 0.771i)7-s + (0.249 + 0.249i)8-s + (0.288 − 0.166i)9-s + (−0.688 − 0.162i)10-s + 1.66·11-s + (0.204 − 0.204i)12-s + (0.191 − 0.716i)13-s + (−0.385 + 0.668i)14-s + (−0.132 + 0.561i)15-s + (0.125 − 0.216i)16-s + (−1.23 + 0.330i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 + 0.368i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.929 + 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.929 + 0.368i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ -0.929 + 0.368i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.156534 - 0.818751i\)
\(L(\frac12)\) \(\approx\) \(0.156534 - 0.818751i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 + 0.965i)T \)
3 \( 1 + (0.965 - 0.258i)T \)
5 \( 1 + (-1.05 + 1.96i)T \)
19 \( 1 + (1.30 - 4.15i)T \)
good7 \( 1 + (2.04 + 2.04i)T + 7iT^{2} \)
11 \( 1 - 5.53T + 11T^{2} \)
13 \( 1 + (-0.692 + 2.58i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + (5.09 - 1.36i)T + (14.7 - 8.5i)T^{2} \)
23 \( 1 + (5.30 + 1.42i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (2.57 + 4.46i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 10.5iT - 31T^{2} \)
37 \( 1 + (-4.21 + 4.21i)T - 37iT^{2} \)
41 \( 1 + (7.73 + 4.46i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.41 - 5.27i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (1.84 - 6.87i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-1.75 + 6.55i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (4.47 - 7.75i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.35 - 5.81i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.309 - 0.0830i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-9.99 - 5.76i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (2.81 + 10.4i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (0.715 - 1.23i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.42 - 2.42i)T - 83iT^{2} \)
89 \( 1 + (-1.87 - 3.25i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.37 + 5.14i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.19798274777817573005852117122, −9.695186407363315482816309406518, −8.896923618549135424634364016906, −7.85692508053207392342580120274, −6.43053047853682206123900143127, −5.87364562645116814707948647175, −4.24785579051749870017543428696, −3.91411209367258229325758235986, −1.91876678489732064822866639808, −0.54238249621307649545830062996, 1.88999747385909506171444574228, 3.48778217736690843051618269310, 4.80291902354362980024511120055, 6.06925754091855622337384351478, 6.64499862172587564547856160960, 6.98788969194982698080605044286, 8.734665952027153637077531841298, 9.237718738893720748200537294975, 10.08459177173433171356195495499, 11.19572377424894253256602195926

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