Properties

Label 2-1960-7.2-c1-0-31
Degree $2$
Conductor $1960$
Sign $-0.266 + 0.963i$
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  + (0.5 − 0.866i)5-s + (1.5 − 2.59i)9-s + (−2 − 3.46i)11-s + 2·13-s + (1 + 1.73i)17-s + (2 − 3.46i)19-s + (−2 + 3.46i)23-s + (−0.499 − 0.866i)25-s − 2·29-s + (−4 − 6.92i)31-s + (−3 + 5.19i)37-s + 6·41-s − 8·43-s + (−1.5 − 2.59i)45-s + (2 − 3.46i)47-s + ⋯
L(s)  = 1  + (0.223 − 0.387i)5-s + (0.5 − 0.866i)9-s + (−0.603 − 1.04i)11-s + 0.554·13-s + (0.242 + 0.420i)17-s + (0.458 − 0.794i)19-s + (−0.417 + 0.722i)23-s + (−0.0999 − 0.173i)25-s − 0.371·29-s + (−0.718 − 1.24i)31-s + (−0.493 + 0.854i)37-s + 0.937·41-s − 1.21·43-s + (−0.223 − 0.387i)45-s + (0.291 − 0.505i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1960\)    =    \(2^{3} \cdot 5 \cdot 7^{2}\)
Sign: $-0.266 + 0.963i$
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: \((2,\ 1960,\ (\ :1/2),\ -0.266 + 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.526149348\)
\(L(\frac12)\) \(\approx\) \(1.526149348\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.5 + 2.59i)T^{2} \)
11 \( 1 + (2 + 3.46i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + (-1 - 1.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2 + 3.46i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + (4 + 6.92i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3 - 5.19i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 + (-2 + 3.46i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3 + 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2 + 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1 - 1.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4 + 6.92i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (3 + 5.19i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 16T + 83T^{2} \)
89 \( 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 14T + 97T^{2} \)
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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

−9.031498475158104799975340742924, −8.202510110926725375561992325429, −7.48000776587592498954195944030, −6.42879345535061633993076438240, −5.81719052051386365643483156906, −4.98457729243109005876415650039, −3.84035859024125448052399271146, −3.18034617942723019355588467044, −1.75196843018542689560959330384, −0.55182041369798724864729292085, 1.55673831530387375016807690684, 2.44272373468884620665589966239, 3.57719903384142266783470345619, 4.61450515674435707832336479544, 5.34286416306667222857382590396, 6.23736011171599227422803098777, 7.31993189195403858431227493458, 7.58908477500992717469597102699, 8.610886997194145311513936571838, 9.512509086409007994572809483649

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