Properties

Label 2-1960-7.2-c1-0-8
Degree $2$
Conductor $1960$
Sign $-0.827 - 0.561i$
Analytic cond. $15.6506$
Root an. cond. $3.95609$
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  + (1.43 + 2.49i)3-s + (0.5 − 0.866i)5-s + (−2.64 + 4.58i)9-s + (−0.732 − 1.26i)11-s − 2.22·13-s + 2.87·15-s + (3.63 + 6.29i)17-s + (−2.74 + 4.75i)19-s + (−1.25 + 2.17i)23-s + (−0.499 − 0.866i)25-s − 6.60·27-s − 7.12·29-s + (−3.25 − 5.64i)31-s + (2.11 − 3.65i)33-s + (−3.45 + 5.97i)37-s + ⋯
L(s)  = 1  + (0.831 + 1.43i)3-s + (0.223 − 0.387i)5-s + (−0.882 + 1.52i)9-s + (−0.220 − 0.382i)11-s − 0.616·13-s + 0.743·15-s + (0.881 + 1.52i)17-s + (−0.629 + 1.09i)19-s + (−0.262 + 0.454i)23-s + (−0.0999 − 0.173i)25-s − 1.27·27-s − 1.32·29-s + (−0.585 − 1.01i)31-s + (0.367 − 0.636i)33-s + (−0.567 + 0.982i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.827 - 0.561i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.827 - 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1960\)    =    \(2^{3} \cdot 5 \cdot 7^{2}\)
Sign: $-0.827 - 0.561i$
Analytic conductor: \(15.6506\)
Root analytic conductor: \(3.95609\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1960} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1960,\ (\ :1/2),\ -0.827 - 0.561i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.858999835\)
\(L(\frac12)\) \(\approx\) \(1.858999835\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 \)
good3 \( 1 + (-1.43 - 2.49i)T + (-1.5 + 2.59i)T^{2} \)
11 \( 1 + (0.732 + 1.26i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 2.22T + 13T^{2} \)
17 \( 1 + (-3.63 - 6.29i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.74 - 4.75i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.25 - 2.17i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 7.12T + 29T^{2} \)
31 \( 1 + (3.25 + 5.64i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.45 - 5.97i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 11.3T + 41T^{2} \)
43 \( 1 - 3.31T + 43T^{2} \)
47 \( 1 + (4.18 - 7.24i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.50 - 6.06i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.53 - 7.85i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.60 + 9.71i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.20 - 5.55i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 10.5T + 71T^{2} \)
73 \( 1 + (-5.26 - 9.11i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.09 - 8.82i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 16.4T + 83T^{2} \)
89 \( 1 + (-4.91 + 8.51i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 2.09T + 97T^{2} \)
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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

−9.667035199134589925463668829131, −8.749033719349428661541614552428, −8.178418086882929996700673352276, −7.50327837793285318162755531849, −5.87222003672502167205443699102, −5.55972354923698549158449667511, −4.27004472867601051292334015579, −3.90197087621052174601491268932, −2.89502611656109036023381537583, −1.75193383268404646405830099782, 0.56365724433097074802322251908, 2.03350911094861567683376027530, 2.56183932602477452443813896045, 3.51415206954428380749684823314, 4.90931086256637607199112421468, 5.82050271476485924295853350166, 7.00035501567745752147911545773, 7.15608854950149485769783020872, 7.85438680069631822599867255241, 8.868563072835522965689784928186

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