Properties

Label 2-570-285.188-c1-0-7
Degree $2$
Conductor $570$
Sign $0.772 - 0.635i$
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  + (−0.819 + 0.573i)2-s + (−1.56 − 0.749i)3-s + (0.342 − 0.939i)4-s + (2.02 + 0.941i)5-s + (1.70 − 0.281i)6-s + (5.01 + 1.34i)7-s + (0.258 + 0.965i)8-s + (1.87 + 2.34i)9-s + (−2.20 + 0.392i)10-s + (−1.73 − 1.00i)11-s + (−1.23 + 1.21i)12-s + (−1.02 − 0.0896i)13-s + (−4.88 + 1.77i)14-s + (−2.46 − 2.99i)15-s + (−0.766 − 0.642i)16-s + (0.388 − 0.272i)17-s + ⋯
L(s)  = 1  + (−0.579 + 0.405i)2-s + (−0.901 − 0.432i)3-s + (0.171 − 0.469i)4-s + (0.907 + 0.421i)5-s + (0.697 − 0.115i)6-s + (1.89 + 0.508i)7-s + (0.0915 + 0.341i)8-s + (0.625 + 0.780i)9-s + (−0.696 + 0.124i)10-s + (−0.524 − 0.302i)11-s + (−0.357 + 0.349i)12-s + (−0.284 − 0.0248i)13-s + (−1.30 + 0.474i)14-s + (−0.635 − 0.772i)15-s + (−0.191 − 0.160i)16-s + (0.0942 − 0.0659i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.772 - 0.635i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.772 - 0.635i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.772 - 0.635i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (473, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.772 - 0.635i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.07719 + 0.386363i\)
\(L(\frac12)\) \(\approx\) \(1.07719 + 0.386363i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.819 - 0.573i)T \)
3 \( 1 + (1.56 + 0.749i)T \)
5 \( 1 + (-2.02 - 0.941i)T \)
19 \( 1 + (4.32 - 0.578i)T \)
good7 \( 1 + (-5.01 - 1.34i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (1.73 + 1.00i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.02 + 0.0896i)T + (12.8 + 2.25i)T^{2} \)
17 \( 1 + (-0.388 + 0.272i)T + (5.81 - 15.9i)T^{2} \)
23 \( 1 + (-5.18 + 2.41i)T + (14.7 - 17.6i)T^{2} \)
29 \( 1 + (-1.24 - 7.06i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (-2.75 - 4.76i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.06 + 2.06i)T - 37iT^{2} \)
41 \( 1 + (4.35 - 5.19i)T + (-7.11 - 40.3i)T^{2} \)
43 \( 1 + (4.07 + 1.90i)T + (27.6 + 32.9i)T^{2} \)
47 \( 1 + (-4.58 + 6.55i)T + (-16.0 - 44.1i)T^{2} \)
53 \( 1 + (-2.69 + 1.25i)T + (34.0 - 40.6i)T^{2} \)
59 \( 1 + (-1.86 + 10.5i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (7.08 + 2.57i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-11.1 - 7.80i)T + (22.9 + 62.9i)T^{2} \)
71 \( 1 + (3.08 + 8.47i)T + (-54.3 + 45.6i)T^{2} \)
73 \( 1 + (-0.271 - 3.10i)T + (-71.8 + 12.6i)T^{2} \)
79 \( 1 + (4.64 - 5.53i)T + (-13.7 - 77.7i)T^{2} \)
83 \( 1 + (0.322 + 0.0864i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + (-5.13 + 4.30i)T + (15.4 - 87.6i)T^{2} \)
97 \( 1 + (-2.88 - 4.12i)T + (-33.1 + 91.1i)T^{2} \)
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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

−10.76993685954004533257505504828, −10.26079259460668928798896510744, −8.850850279455631818423415951565, −8.190925557222968858228105337755, −7.17744302202182493960795957970, −6.37370292747867741012301767068, −5.23159490340484296520946638965, −4.97075371866219133170410440690, −2.37695284740573607895057686373, −1.38514001989258606692847261427, 1.05801127821729469186822477378, 2.22318770272253616765072671685, 4.32832532154231503630795094518, 4.90048830264084186712350590368, 5.90762292280125085000904286901, 7.17452369018221808119579004021, 8.106241663087090308199697177501, 9.032129301415798811469780829535, 10.03365258194003224211661660695, 10.58929132807017818298937794957

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