Properties

Label 2-2070-15.8-c1-0-32
Degree $2$
Conductor $2070$
Sign $0.927 - 0.374i$
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 + (2.12 − 0.707i)5-s + (−0.707 + 0.707i)8-s + (2 + 0.999i)10-s − 2.82i·11-s + (−3 − 3i)13-s − 1.00·16-s + (2.82 + 2.82i)17-s + 4i·19-s + (0.707 + 2.12i)20-s + (2.00 − 2.00i)22-s + (−0.707 + 0.707i)23-s + (3.99 − 3i)25-s − 4.24i·26-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + 0.500i·4-s + (0.948 − 0.316i)5-s + (−0.250 + 0.250i)8-s + (0.632 + 0.316i)10-s − 0.852i·11-s + (−0.832 − 0.832i)13-s − 0.250·16-s + (0.685 + 0.685i)17-s + 0.917i·19-s + (0.158 + 0.474i)20-s + (0.426 − 0.426i)22-s + (−0.147 + 0.147i)23-s + (0.799 − 0.600i)25-s − 0.832i·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 - 0.374i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.927 - 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2070\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $0.927 - 0.374i$
Analytic conductor: \(16.5290\)
Root analytic conductor: \(4.06559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2070} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2070,\ (\ :1/2),\ 0.927 - 0.374i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.804246128\)
\(L(\frac12)\) \(\approx\) \(2.804246128\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.12 + 0.707i)T \)
23 \( 1 + (0.707 - 0.707i)T \)
good7 \( 1 - 7iT^{2} \)
11 \( 1 + 2.82iT - 11T^{2} \)
13 \( 1 + (3 + 3i)T + 13iT^{2} \)
17 \( 1 + (-2.82 - 2.82i)T + 17iT^{2} \)
19 \( 1 - 4iT - 19T^{2} \)
29 \( 1 - 9.89T + 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 + (-7 + 7i)T - 37iT^{2} \)
41 \( 1 - 7.07iT - 41T^{2} \)
43 \( 1 + (-2 - 2i)T + 43iT^{2} \)
47 \( 1 + (2.82 + 2.82i)T + 47iT^{2} \)
53 \( 1 + (-8.48 + 8.48i)T - 53iT^{2} \)
59 \( 1 + 11.3T + 59T^{2} \)
61 \( 1 + 4T + 61T^{2} \)
67 \( 1 + (-10 + 10i)T - 67iT^{2} \)
71 \( 1 + 14.1iT - 71T^{2} \)
73 \( 1 + (-1 - i)T + 73iT^{2} \)
79 \( 1 + 4iT - 79T^{2} \)
83 \( 1 + (8.48 - 8.48i)T - 83iT^{2} \)
89 \( 1 + 9.89T + 89T^{2} \)
97 \( 1 + (-3 + 3i)T - 97iT^{2} \)
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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

−9.099118695181456932068518202522, −8.099400030953032235504033687247, −7.85801542369700998923062862754, −6.41765892315836364799264553632, −6.08029868518721278892245931567, −5.26542487366916744802791702977, −4.51448506179189041368230527320, −3.31014646632361881496256017078, −2.49340188267038863365719764576, −1.02986406041960055157200638413, 1.16055218300553806200644341881, 2.44345064198214491609156656820, 2.83072815466402933564986921391, 4.41233228292337686585482992965, 4.83964477779042021106035940591, 5.81619907731749240020066906539, 6.75168062269913017640962593562, 7.16791969622481918160130029003, 8.453247462135531114001097686862, 9.395920775107525535068607589526

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