Properties

Label 2-2070-15.2-c1-0-11
Degree $2$
Conductor $2070$
Sign $0.662 - 0.749i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2070\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $0.662 - 0.749i$
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: \((2,\ 2070,\ (\ :1/2),\ 0.662 - 0.749i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8013015653\)
\(L(\frac12)\) \(\approx\) \(0.8013015653\)
\(L(\frac{3}{2})\) 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

−9.105851463025712865115265874971, −8.356202093857425311345808977896, −7.82336593076467849371328939962, −6.90611795787722322229822338776, −6.34542991958962523680944798431, −5.16352437927969192216389878360, −4.48836831684164398197780786827, −3.52077737151280507137087883588, −2.23126284303152791430322724905, −0.73851529732837011238998420501, 0.52350747622087764878506862670, 2.15080119458826458329038494823, 2.98343336744808391698613525032, 4.03632283175555575849766188015, 4.70969786663937874919580518670, 5.87609969980516257166134290198, 7.06570378659244795528854079048, 7.56441136569945313064585197752, 8.105323519522208816773790658135, 9.180709176071292907301366047830

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