Properties

Label 2-570-285.29-c1-0-10
Degree $2$
Conductor $570$
Sign $0.971 + 0.235i$
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  + (−0.984 + 0.173i)2-s + (−1.70 − 0.302i)3-s + (0.939 − 0.342i)4-s + (0.247 − 2.22i)5-s + (1.73 + 0.00201i)6-s + (4.20 + 2.43i)7-s + (−0.866 + 0.5i)8-s + (2.81 + 1.03i)9-s + (0.142 + 2.23i)10-s + (−0.466 + 0.269i)11-s + (−1.70 + 0.298i)12-s + (−3.83 − 3.21i)13-s + (−4.56 − 1.66i)14-s + (−1.09 + 3.71i)15-s + (0.766 − 0.642i)16-s + (0.787 + 4.46i)17-s + ⋯
L(s)  = 1  + (−0.696 + 0.122i)2-s + (−0.984 − 0.174i)3-s + (0.469 − 0.171i)4-s + (0.110 − 0.993i)5-s + (0.707 + 0.000823i)6-s + (1.59 + 0.918i)7-s + (−0.306 + 0.176i)8-s + (0.938 + 0.344i)9-s + (0.0450 + 0.705i)10-s + (−0.140 + 0.0812i)11-s + (−0.492 + 0.0862i)12-s + (−1.06 − 0.892i)13-s + (−1.22 − 0.444i)14-s + (−0.282 + 0.959i)15-s + (0.191 − 0.160i)16-s + (0.191 + 1.08i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 + 0.235i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.971 + 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.971 + 0.235i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.971 + 0.235i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.918737 - 0.109902i\)
\(L(\frac12)\) \(\approx\) \(0.918737 - 0.109902i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.984 - 0.173i)T \)
3 \( 1 + (1.70 + 0.302i)T \)
5 \( 1 + (-0.247 + 2.22i)T \)
19 \( 1 + (-2.88 - 3.26i)T \)
good7 \( 1 + (-4.20 - 2.43i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.466 - 0.269i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.83 + 3.21i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (-0.787 - 4.46i)T + (-15.9 + 5.81i)T^{2} \)
23 \( 1 + (-3.77 + 1.37i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (-0.161 + 0.913i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (-4.96 - 2.86i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 8.71T + 37T^{2} \)
41 \( 1 + (-0.823 + 0.690i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-4.14 + 11.3i)T + (-32.9 - 27.6i)T^{2} \)
47 \( 1 + (-1.74 + 9.90i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (-3.29 - 9.05i)T + (-40.6 + 34.0i)T^{2} \)
59 \( 1 + (0.0388 + 0.220i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (2.38 - 0.866i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-0.877 + 4.97i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-3.58 - 1.30i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (-0.384 - 0.458i)T + (-12.6 + 71.8i)T^{2} \)
79 \( 1 + (3.29 + 3.92i)T + (-13.7 + 77.7i)T^{2} \)
83 \( 1 + (3.89 - 6.73i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (3.04 + 2.55i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (1.78 + 10.1i)T + (-91.1 + 33.1i)T^{2} \)
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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

−10.64603843513537185848942935411, −9.940418546995651820960053773132, −8.793203830369999172914386657292, −8.063810572103947334975045460838, −7.41158578466160056738372652872, −5.83083854546036231086457219195, −5.36928644641252037964014391952, −4.52062526715112787325228921181, −2.16873923219519196191756796106, −1.04445880944137967080149817637, 1.07070595780907460791022590593, 2.61372994219573150237610062939, 4.35178385064480014917169418778, 5.10269791903354780245539975934, 6.49662503540449150234401160472, 7.37249618040415092280217093439, 7.69668239389712589457889518140, 9.394996282574481702861087559201, 9.960756607159430595020788000643, 10.99794941641400404189273573827

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