Properties

Label 2-570-285.263-c1-0-7
Degree $2$
Conductor $570$
Sign $0.237 - 0.971i$
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 + (0.363 − 1.69i)3-s + (−0.984 + 0.173i)4-s + (−1.95 + 1.08i)5-s + (1.71 + 0.214i)6-s + (−1.28 − 0.343i)7-s + (−0.258 − 0.965i)8-s + (−2.73 − 1.23i)9-s + (−1.25 − 1.85i)10-s + (4.64 + 2.68i)11-s + (−0.0638 + 1.73i)12-s + (1.47 + 3.17i)13-s + (0.230 − 1.30i)14-s + (1.13 + 3.70i)15-s + (0.939 − 0.342i)16-s + (0.458 + 5.23i)17-s + ⋯
L(s)  = 1  + (0.0616 + 0.704i)2-s + (0.209 − 0.977i)3-s + (−0.492 + 0.0868i)4-s + (−0.873 + 0.487i)5-s + (0.701 + 0.0875i)6-s + (−0.485 − 0.129i)7-s + (−0.0915 − 0.341i)8-s + (−0.911 − 0.410i)9-s + (−0.397 − 0.585i)10-s + (1.40 + 0.808i)11-s + (−0.0184 + 0.499i)12-s + (0.409 + 0.879i)13-s + (0.0616 − 0.349i)14-s + (0.293 + 0.956i)15-s + (0.234 − 0.0855i)16-s + (0.111 + 1.27i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.237 - 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.237 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.237 - 0.971i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.237 - 0.971i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.941306 + 0.739071i\)
\(L(\frac12)\) \(\approx\) \(0.941306 + 0.739071i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.363 + 1.69i)T \)
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

−11.29528102537757393947790317367, −9.738331685023193807332730147228, −8.981059539416203650317409166761, −8.042523901358925156732808480157, −7.07249974699013811280737861031, −6.80317785706653274535539891760, −5.76711659441602295821155820429, −4.11713861491859195557902796635, −3.38038630189954605527846063381, −1.47629264688173619658619543809, 0.74109116647114220454006093336, 3.13930885854470970304245395500, 3.52345639355318865767612409676, 4.73552074099631517800209961367, 5.54140806594251869882592384367, 7.01672191184952619591396372932, 8.332611718321124640579215792082, 9.052900060404188631129574088100, 9.498824426752459739874659391253, 10.77635548865824875302193927193

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