Properties

Label 2-2070-15.8-c1-0-2
Degree $2$
Conductor $2070$
Sign $-0.510 + 0.859i$
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 + (−1.25 + 1.84i)5-s + (−0.707 + 0.707i)8-s + (−2.19 + 0.417i)10-s + 6.26i·11-s + (−4.39 − 4.39i)13-s − 1.00·16-s + (−2.28 − 2.28i)17-s + 0.722i·19-s + (−1.84 − 1.25i)20-s + (−4.42 + 4.42i)22-s + (−0.707 + 0.707i)23-s + (−1.83 − 4.65i)25-s − 6.21i·26-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + 0.500i·4-s + (−0.562 + 0.826i)5-s + (−0.250 + 0.250i)8-s + (−0.694 + 0.132i)10-s + 1.88i·11-s + (−1.21 − 1.21i)13-s − 0.250·16-s + (−0.553 − 0.553i)17-s + 0.165i·19-s + (−0.413 − 0.281i)20-s + (−0.944 + 0.944i)22-s + (−0.147 + 0.147i)23-s + (−0.367 − 0.930i)25-s − 1.21i·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.510 + 0.859i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.510 + 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2070\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $-0.510 + 0.859i$
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: \((2,\ 2070,\ (\ :1/2),\ -0.510 + 0.859i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3701817742\)
\(L(\frac12)\) \(\approx\) \(0.3701817742\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.25 - 1.84i)T \)
23 \( 1 + (0.707 - 0.707i)T \)
good7 \( 1 - 7iT^{2} \)
11 \( 1 - 6.26iT - 11T^{2} \)
13 \( 1 + (4.39 + 4.39i)T + 13iT^{2} \)
17 \( 1 + (2.28 + 2.28i)T + 17iT^{2} \)
19 \( 1 - 0.722iT - 19T^{2} \)
29 \( 1 - 5.95T + 29T^{2} \)
31 \( 1 + 6T + 31T^{2} \)
37 \( 1 + (-1.05 + 1.05i)T - 37iT^{2} \)
41 \( 1 + 6.80iT - 41T^{2} \)
43 \( 1 + (-1.20 - 1.20i)T + 43iT^{2} \)
47 \( 1 + (8.78 + 8.78i)T + 47iT^{2} \)
53 \( 1 + (2.23 - 2.23i)T - 53iT^{2} \)
59 \( 1 - 12.7T + 59T^{2} \)
61 \( 1 - 4.90T + 61T^{2} \)
67 \( 1 + (7.15 - 7.15i)T - 67iT^{2} \)
71 \( 1 - 3.68iT - 71T^{2} \)
73 \( 1 + (11.6 + 11.6i)T + 73iT^{2} \)
79 \( 1 - 0.442iT - 79T^{2} \)
83 \( 1 + (11.9 - 11.9i)T - 83iT^{2} \)
89 \( 1 + 5.95T + 89T^{2} \)
97 \( 1 + (-6.97 + 6.97i)T - 97iT^{2} \)
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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

−9.884096438183772477962017292711, −8.677920192107306677621812415207, −7.69461702780691572321689022649, −7.25097019501879534336291433465, −6.78138794560370870402913515593, −5.58578506291444627166381720482, −4.79271169946250851928965454556, −4.08979637935471070499078977535, −2.95659758246171932118936944117, −2.20902422635025912448030910039, 0.10660761960426614525355710119, 1.43089226952879790182066216369, 2.69155319331269731288877005045, 3.69403845814619855395787004696, 4.46971159354111376194940720493, 5.17771001333524734834877253909, 6.11487224773346195586244485799, 6.90534612203940278607801227996, 8.028001964578716124049762172556, 8.673985212935900510601749558426

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