Properties

Label 2-2070-15.2-c1-0-36
Degree $2$
Conductor $2070$
Sign $-0.487 + 0.873i$
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 + (0.967 − 2.01i)5-s + (0.545 + 0.545i)7-s + (−0.707 − 0.707i)8-s + (−0.741 − 2.10i)10-s + 0.823i·11-s + (3.43 − 3.43i)13-s + 0.771·14-s − 1.00·16-s + (−0.823 + 0.823i)17-s − 3.38i·19-s + (−2.01 − 0.967i)20-s + (0.582 + 0.582i)22-s + (0.707 + 0.707i)23-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s − 0.500i·4-s + (0.432 − 0.901i)5-s + (0.206 + 0.206i)7-s + (−0.250 − 0.250i)8-s + (−0.234 − 0.667i)10-s + 0.248i·11-s + (0.952 − 0.952i)13-s + 0.206·14-s − 0.250·16-s + (−0.199 + 0.199i)17-s − 0.776i·19-s + (−0.450 − 0.216i)20-s + (0.124 + 0.124i)22-s + (0.147 + 0.147i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.487 + 0.873i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.487 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2070\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $-0.487 + 0.873i$
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: \((2,\ 2070,\ (\ :1/2),\ -0.487 + 0.873i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.500778894\)
\(L(\frac12)\) \(\approx\) \(2.500778894\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.967 + 2.01i)T \)
23 \( 1 + (-0.707 - 0.707i)T \)
good7 \( 1 + (-0.545 - 0.545i)T + 7iT^{2} \)
11 \( 1 - 0.823iT - 11T^{2} \)
13 \( 1 + (-3.43 + 3.43i)T - 13iT^{2} \)
17 \( 1 + (0.823 - 0.823i)T - 17iT^{2} \)
19 \( 1 + 3.38iT - 19T^{2} \)
29 \( 1 - 9.51T + 29T^{2} \)
31 \( 1 + 6.25T + 31T^{2} \)
37 \( 1 + (2.43 + 2.43i)T + 37iT^{2} \)
41 \( 1 + 0.0889iT - 41T^{2} \)
43 \( 1 + (-1.12 + 1.12i)T - 43iT^{2} \)
47 \( 1 + (-1.57 + 1.57i)T - 47iT^{2} \)
53 \( 1 + (-3.37 - 3.37i)T + 53iT^{2} \)
59 \( 1 + 13.3T + 59T^{2} \)
61 \( 1 - 15.2T + 61T^{2} \)
67 \( 1 + (6.25 + 6.25i)T + 67iT^{2} \)
71 \( 1 - 0.428iT - 71T^{2} \)
73 \( 1 + (7.35 - 7.35i)T - 73iT^{2} \)
79 \( 1 + 1.82iT - 79T^{2} \)
83 \( 1 + (-3.05 - 3.05i)T + 83iT^{2} \)
89 \( 1 + 5.58T + 89T^{2} \)
97 \( 1 + (9.10 + 9.10i)T + 97iT^{2} \)
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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

−8.781819075416466615773189713162, −8.422255452159452500462345920431, −7.25029806279935065006292198274, −6.22182734051787228685884853206, −5.50879374197498517799331186292, −4.85353672935126741081768379239, −3.97781597710835633671251075391, −2.91496940891863336079246346449, −1.84160743202111642181849471382, −0.77389918798316984760982503391, 1.56577675405695738743597345596, 2.77160157718379983334162371630, 3.67550724568073933393136171370, 4.49047603926187156876904369861, 5.57448767463326652023268915238, 6.31249696086672182804883426664, 6.82868068151041639024132323751, 7.66253397906490855690465128629, 8.529552354646098919788285305845, 9.246314113229639908738645985949

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