Properties

Label 2-570-285.272-c1-0-10
Degree $2$
Conductor $570$
Sign $-0.636 - 0.771i$
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  + (−0.0871 + 0.996i)2-s + (−1.06 − 1.36i)3-s + (−0.984 − 0.173i)4-s + (1.95 + 1.08i)5-s + (1.45 − 0.940i)6-s + (−1.28 + 0.343i)7-s + (0.258 − 0.965i)8-s + (−0.737 + 2.90i)9-s + (−1.25 + 1.85i)10-s + (−4.64 + 2.68i)11-s + (0.810 + 1.53i)12-s + (1.47 − 3.17i)13-s + (−0.230 − 1.30i)14-s + (−0.586 − 3.82i)15-s + (0.939 + 0.342i)16-s + (−0.458 + 5.23i)17-s + ⋯
L(s)  = 1  + (−0.0616 + 0.704i)2-s + (−0.614 − 0.789i)3-s + (−0.492 − 0.0868i)4-s + (0.873 + 0.487i)5-s + (0.593 − 0.383i)6-s + (−0.485 + 0.129i)7-s + (0.0915 − 0.341i)8-s + (−0.245 + 0.969i)9-s + (−0.397 + 0.585i)10-s + (−1.40 + 0.808i)11-s + (0.233 + 0.441i)12-s + (0.409 − 0.879i)13-s + (−0.0616 − 0.349i)14-s + (−0.151 − 0.988i)15-s + (0.234 + 0.0855i)16-s + (−0.111 + 1.27i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.636 - 0.771i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.636 - 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.636 - 0.771i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ -0.636 - 0.771i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.324391 + 0.687792i\)
\(L(\frac12)\) \(\approx\) \(0.324391 + 0.687792i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0871 - 0.996i)T \)
3 \( 1 + (1.06 + 1.36i)T \)
5 \( 1 + (-1.95 - 1.08i)T \)
19 \( 1 + (-4.21 + 1.12i)T \)
good7 \( 1 + (1.28 - 0.343i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (4.64 - 2.68i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.47 + 3.17i)T + (-8.35 - 9.95i)T^{2} \)
17 \( 1 + (0.458 - 5.23i)T + (-16.7 - 2.95i)T^{2} \)
23 \( 1 + (6.24 - 4.37i)T + (7.86 - 21.6i)T^{2} \)
29 \( 1 + (-1.96 - 1.64i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (4.70 - 8.15i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.77 + 2.77i)T + 37iT^{2} \)
41 \( 1 + (1.57 - 4.31i)T + (-31.4 - 26.3i)T^{2} \)
43 \( 1 + (-0.480 - 0.336i)T + (14.7 + 40.4i)T^{2} \)
47 \( 1 + (-3.09 + 0.270i)T + (46.2 - 8.16i)T^{2} \)
53 \( 1 + (-1.61 + 1.12i)T + (18.1 - 49.8i)T^{2} \)
59 \( 1 + (11.6 - 9.74i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (1.33 - 7.59i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (1.05 + 12.0i)T + (-65.9 + 11.6i)T^{2} \)
71 \( 1 + (-5.16 + 0.910i)T + (66.7 - 24.2i)T^{2} \)
73 \( 1 + (-11.3 + 5.30i)T + (46.9 - 55.9i)T^{2} \)
79 \( 1 + (-0.195 + 0.538i)T + (-60.5 - 50.7i)T^{2} \)
83 \( 1 + (-12.9 + 3.47i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (-7.24 + 2.63i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (-3.27 - 0.286i)T + (95.5 + 16.8i)T^{2} \)
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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

−10.60208582422165560558831956116, −10.44289381035030374541765566455, −9.304915936724215241432905698095, −8.023797129633726917981746698589, −7.43672160434141110292106126546, −6.44606444808865299501623127299, −5.71535284280088442321661604433, −5.09029792596537169747860328860, −3.15814021405825584401053272489, −1.74567443075808619193196208686, 0.46324460497519967589216818568, 2.40491866687345759496016091393, 3.63581334645433136177677124704, 4.83446340515815782383740846362, 5.57111819176150151348538788748, 6.46217108633073767074263902099, 8.030486436561068051940356540824, 9.119861920170293763672827504346, 9.696389178743851993994474808138, 10.36046907836866659895316956713

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