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Properties

Label 2-2070-15.2-c1-0-39
Degree $2$
Conductor $2070$
Sign $-0.981 + 0.188i$
Analytic cond. $16.5290$
Root an. cond. $4.06559$
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  + (−0.707 + 0.707i)2-s − 1.00i·4-s + (−1.24 − 1.85i)5-s + (−3.35 − 3.35i)7-s + (0.707 + 0.707i)8-s + (2.19 + 0.430i)10-s − 4.58i·11-s + (0.617 − 0.617i)13-s + 4.74·14-s − 1.00·16-s + (4.58 − 4.58i)17-s − 3.62i·19-s + (−1.85 + 1.24i)20-s + (3.24 + 3.24i)22-s + (−0.707 − 0.707i)23-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s − 0.500i·4-s + (−0.557 − 0.829i)5-s + (−1.26 − 1.26i)7-s + (0.250 + 0.250i)8-s + (0.693 + 0.136i)10-s − 1.38i·11-s + (0.171 − 0.171i)13-s + 1.26·14-s − 0.250·16-s + (1.11 − 1.11i)17-s − 0.831i·19-s + (−0.414 + 0.278i)20-s + (0.691 + 0.691i)22-s + (−0.147 − 0.147i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 + 0.188i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.981 + 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2070\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $-0.981 + 0.188i$
Analytic conductor: \(16.5290\)
Root analytic conductor: \(4.06559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2070} (737, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2070,\ (\ :1/2),\ -0.981 + 0.188i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7235889615\)
\(L(\frac12)\) \(\approx\) \(0.7235889615\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 \)
5 \( 1 + (1.24 + 1.85i)T \)
23 \( 1 + (0.707 + 0.707i)T \)
good7 \( 1 + (3.35 + 3.35i)T + 7iT^{2} \)
11 \( 1 + 4.58iT - 11T^{2} \)
13 \( 1 + (-0.617 + 0.617i)T - 13iT^{2} \)
17 \( 1 + (-4.58 + 4.58i)T - 17iT^{2} \)
19 \( 1 + 3.62iT - 19T^{2} \)
29 \( 1 - 9.41T + 29T^{2} \)
31 \( 1 + 3.77T + 31T^{2} \)
37 \( 1 + (4.79 + 4.79i)T + 37iT^{2} \)
41 \( 1 - 2.87iT - 41T^{2} \)
43 \( 1 + (0.112 - 0.112i)T - 43iT^{2} \)
47 \( 1 + (-7.28 + 7.28i)T - 47iT^{2} \)
53 \( 1 + (5.77 + 5.77i)T + 53iT^{2} \)
59 \( 1 + 10.6T + 59T^{2} \)
61 \( 1 - 12.7T + 61T^{2} \)
67 \( 1 + (7.73 + 7.73i)T + 67iT^{2} \)
71 \( 1 + 5.53iT - 71T^{2} \)
73 \( 1 + (8.22 - 8.22i)T - 73iT^{2} \)
79 \( 1 - 5.17iT - 79T^{2} \)
83 \( 1 + (-4.85 - 4.85i)T + 83iT^{2} \)
89 \( 1 - 9.90T + 89T^{2} \)
97 \( 1 + (5.11 + 5.11i)T + 97iT^{2} \)
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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

−8.763967589352068054597078793110, −7.955861500319917078465466020965, −7.25499123709362980194517491485, −6.57683983106113210769238400998, −5.66202308295500280919598613207, −4.80160005638256872681812909176, −3.71705997836411557958901726441, −3.05098189010038601593125731002, −0.912827228057638580955960559651, −0.40660457137342839909106030249, 1.72173821956226840336921710577, 2.77554111620892058857749551363, 3.41569779593365496543741110634, 4.36015675121548644610040107794, 5.74899405016604998032604561600, 6.41424652999227197481856847091, 7.22173065394698530820896101469, 8.020040853758033884332663040869, 8.786710268709574673290791345953, 9.654369974188007448504564722811

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