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Properties

Label 2-570-95.27-c1-0-1
Degree $2$
Conductor $570$
Sign $0.219 - 0.975i$
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 + (−1.23 + 1.86i)5-s + (−0.499 − 0.866i)6-s + (1.93 + 1.93i)7-s + (0.707 + 0.707i)8-s + (0.866 − 0.499i)9-s + (2.11 + 0.715i)10-s − 6.13·11-s + (−0.707 + 0.707i)12-s + (−1.66 + 6.20i)13-s + (1.36 − 2.37i)14-s + (−0.715 + 2.11i)15-s + (0.500 − 0.866i)16-s + (−3.56 + 0.956i)17-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s + (0.557 − 0.149i)3-s + (−0.433 + 0.249i)4-s + (−0.554 + 0.832i)5-s + (−0.204 − 0.353i)6-s + (0.731 + 0.731i)7-s + (0.249 + 0.249i)8-s + (0.288 − 0.166i)9-s + (0.669 + 0.226i)10-s − 1.85·11-s + (−0.204 + 0.204i)12-s + (−0.461 + 1.72i)13-s + (0.365 − 0.633i)14-s + (−0.184 + 0.546i)15-s + (0.125 − 0.216i)16-s + (−0.865 + 0.231i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.219 - 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.219 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.219 - 0.975i$
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: \((2,\ 570,\ (\ :1/2),\ 0.219 - 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.739665 + 0.592043i\)
\(L(\frac12)\) \(\approx\) \(0.739665 + 0.592043i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.23 - 1.86i)T \)
19 \( 1 + (3.01 + 3.15i)T \)
good7 \( 1 + (-1.93 - 1.93i)T + 7iT^{2} \)
11 \( 1 + 6.13T + 11T^{2} \)
13 \( 1 + (1.66 - 6.20i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + (3.56 - 0.956i)T + (14.7 - 8.5i)T^{2} \)
23 \( 1 + (0.234 + 0.0627i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (-3.45 - 5.99i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 7.31iT - 31T^{2} \)
37 \( 1 + (-7.48 + 7.48i)T - 37iT^{2} \)
41 \( 1 + (-4.33 - 2.50i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.46 - 5.48i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (-2.04 + 7.64i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-0.398 + 1.48i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-0.890 + 1.54i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.98 - 10.3i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.33 - 1.96i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (2.59 + 1.50i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-0.428 - 1.59i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-2.06 + 3.56i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.54 - 4.54i)T - 83iT^{2} \)
89 \( 1 + (3.75 + 6.50i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.640 - 2.39i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.98152221774872934438430579827, −10.24342901590050162106873108986, −9.027739938153459989156167795552, −8.452064612393272486357101227381, −7.52292345420272631039365746301, −6.67186919428588561148235811231, −5.05291812599485233373179920451, −4.17618221744988531021986841101, −2.66975943232970669766862137368, −2.19903361072942085495388704134, 0.51177411623893105993304491666, 2.56669774861569707710277398586, 4.17615688953683931748115890997, 4.88253798915867857275025972908, 5.80963516688485890678403092649, 7.47955542911458817869683844141, 8.057915976093262835338297972673, 8.220129542536542446216568769054, 9.642059065146176417438445527726, 10.39714315903116281003970373144

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