Properties

Label 2-570-285.233-c1-0-3
Degree $2$
Conductor $570$
Sign $-0.994 + 0.106i$
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  + (−0.906 − 0.422i)2-s + (−0.0197 + 1.73i)3-s + (0.642 + 0.766i)4-s + (1.82 + 1.28i)5-s + (0.749 − 1.56i)6-s + (−2.23 − 0.599i)7-s + (−0.258 − 0.965i)8-s + (−2.99 − 0.0683i)9-s + (−1.11 − 1.93i)10-s + (−4.47 − 2.58i)11-s + (−1.33 + 1.09i)12-s + (−3.32 + 4.74i)13-s + (1.77 + 1.48i)14-s + (−2.26 + 3.13i)15-s + (−0.173 + 0.984i)16-s + (−3.71 − 1.73i)17-s + ⋯
L(s)  = 1  + (−0.640 − 0.298i)2-s + (−0.0113 + 0.999i)3-s + (0.321 + 0.383i)4-s + (0.817 + 0.576i)5-s + (0.306 − 0.637i)6-s + (−0.845 − 0.226i)7-s + (−0.0915 − 0.341i)8-s + (−0.999 − 0.0227i)9-s + (−0.351 − 0.613i)10-s + (−1.35 − 0.779i)11-s + (−0.386 + 0.317i)12-s + (−0.922 + 1.31i)13-s + (0.473 + 0.397i)14-s + (−0.585 + 0.810i)15-s + (−0.0434 + 0.246i)16-s + (−0.900 − 0.419i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.106i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 + 0.106i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.994 + 0.106i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (233, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ -0.994 + 0.106i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0166936 - 0.313828i\)
\(L(\frac12)\) \(\approx\) \(0.0166936 - 0.313828i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.906 + 0.422i)T \)
3 \( 1 + (0.0197 - 1.73i)T \)
5 \( 1 + (-1.82 - 1.28i)T \)
19 \( 1 + (-4.35 - 0.273i)T \)
good7 \( 1 + (2.23 + 0.599i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (4.47 + 2.58i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.32 - 4.74i)T + (-4.44 - 12.2i)T^{2} \)
17 \( 1 + (3.71 + 1.73i)T + (10.9 + 13.0i)T^{2} \)
23 \( 1 + (-0.253 + 2.89i)T + (-22.6 - 3.99i)T^{2} \)
29 \( 1 + (3.37 + 1.22i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (2.06 + 3.58i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.36 - 4.36i)T - 37iT^{2} \)
41 \( 1 + (-5.60 - 0.988i)T + (38.5 + 14.0i)T^{2} \)
43 \( 1 + (-0.946 - 10.8i)T + (-42.3 + 7.46i)T^{2} \)
47 \( 1 + (-0.232 - 0.499i)T + (-30.2 + 36.0i)T^{2} \)
53 \( 1 + (0.614 - 7.02i)T + (-52.1 - 9.20i)T^{2} \)
59 \( 1 + (3.30 - 1.20i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (4.90 - 4.11i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (-11.0 + 5.13i)T + (43.0 - 51.3i)T^{2} \)
71 \( 1 + (-4.62 + 5.51i)T + (-12.3 - 69.9i)T^{2} \)
73 \( 1 + (7.92 - 5.55i)T + (24.9 - 68.5i)T^{2} \)
79 \( 1 + (0.848 + 0.149i)T + (74.2 + 27.0i)T^{2} \)
83 \( 1 + (16.6 + 4.46i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + (1.20 + 6.84i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (3.47 - 7.44i)T + (-62.3 - 74.3i)T^{2} \)
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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

−10.97837574449056079967394297701, −10.15769682794778392065941290781, −9.553945354764967362784544774535, −9.037382452110524087410899806797, −7.72198045106208597413631157284, −6.69195614870557220954360350847, −5.74730085544910474249740376934, −4.58441841464379036372873876719, −3.15071662514281414727911908424, −2.45209290466828035991471983903, 0.19672044249549570281188183428, 1.95344243579446260724588829529, 2.89140674671998468535283216980, 5.30467985107675924768063394954, 5.59949736485088813003335913344, 6.90885409819773047522108427826, 7.53262094780182298950522696676, 8.458382764529517238985336447105, 9.394126778233976451428445631222, 10.07991992003786192280878261173

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