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Properties

Label 2-1960-280.229-c0-0-3
Degree $2$
Conductor $1960$
Sign $0.980 + 0.197i$
Analytic cond. $0.978167$
Root an. cond. $0.989023$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 + 0.5i)2-s + (1.22 − 0.707i)3-s + (0.499 + 0.866i)4-s + (−0.258 − 0.965i)5-s + 1.41·6-s + 0.999i·8-s + (0.499 − 0.866i)9-s + (0.258 − 0.965i)10-s + (1.22 + 0.707i)12-s − 1.41i·13-s + (−1 − 0.999i)15-s + (−0.5 + 0.866i)16-s + (0.866 − 0.5i)18-s + (−0.707 + 1.22i)19-s + (0.707 − 0.707i)20-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (1.22 − 0.707i)3-s + (0.499 + 0.866i)4-s + (−0.258 − 0.965i)5-s + 1.41·6-s + 0.999i·8-s + (0.499 − 0.866i)9-s + (0.258 − 0.965i)10-s + (1.22 + 0.707i)12-s − 1.41i·13-s + (−1 − 0.999i)15-s + (−0.5 + 0.866i)16-s + (0.866 − 0.5i)18-s + (−0.707 + 1.22i)19-s + (0.707 − 0.707i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.197i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.197i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1960\)    =    \(2^{3} \cdot 5 \cdot 7^{2}\)
Sign: $0.980 + 0.197i$
Analytic conductor: \(0.978167\)
Root analytic conductor: \(0.989023\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1960} (509, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1960,\ (\ :0),\ 0.980 + 0.197i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.617922195\)
\(L(\frac12)\) \(\approx\) \(2.617922195\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
5 \( 1 + (0.258 + 0.965i)T \)
7 \( 1 \)
good3 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + 1.41iT - T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + 1.41iT - T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + T^{2} \)
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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

−8.795455517408715508733960642210, −8.405251267885107520142154893168, −7.78138399162079739388607516529, −7.25817110873785222789932263928, −6.05762581781152762638045095454, −5.40537783583394701166883426506, −4.33689639881926973898823755686, −3.52616595254925240264787790112, −2.67145619408963139624900722655, −1.56536576259834493231071747635, 2.05752412513748098901810135145, 2.68997863734978589896465065946, 3.60048503951555610733294106902, 4.17898005187415460072210819400, 4.97173683983579520179917165342, 6.37925232679492813803798751673, 6.83491338672865955891412447169, 7.82628332952882187833173460238, 8.877608814094367311758564973325, 9.470565237649970686026841060937

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