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Properties

Label 2-1960-5.4-c1-0-15
Degree $2$
Conductor $1960$
Sign $-0.141 - 0.989i$
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  + 0.836i·3-s + (−2.21 + 0.317i)5-s + 2.30·9-s + 0.403·11-s + 6.20i·13-s + (−0.265 − 1.85i)15-s − 2.62i·17-s + 6.41·19-s − 6.54i·23-s + (4.79 − 1.40i)25-s + 4.43i·27-s − 1.96·29-s − 4.66·31-s + 0.337i·33-s − 3.44i·37-s + ⋯
L(s)  = 1  + 0.483i·3-s + (−0.989 + 0.141i)5-s + 0.766·9-s + 0.121·11-s + 1.72i·13-s + (−0.0684 − 0.478i)15-s − 0.636i·17-s + 1.47·19-s − 1.36i·23-s + (0.959 − 0.280i)25-s + 0.853i·27-s − 0.364·29-s − 0.838·31-s + 0.0587i·33-s − 0.566i·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.141 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.141 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1960\)    =    \(2^{3} \cdot 5 \cdot 7^{2}\)
Sign: $-0.141 - 0.989i$
Analytic conductor: \(15.6506\)
Root analytic conductor: \(3.95609\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1960} (1569, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1960,\ (\ :1/2),\ -0.141 - 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.364945608\)
\(L(\frac12)\) \(\approx\) \(1.364945608\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.21 - 0.317i)T \)
7 \( 1 \)
good3 \( 1 - 0.836iT - 3T^{2} \)
11 \( 1 - 0.403T + 11T^{2} \)
13 \( 1 - 6.20iT - 13T^{2} \)
17 \( 1 + 2.62iT - 17T^{2} \)
19 \( 1 - 6.41T + 19T^{2} \)
23 \( 1 + 6.54iT - 23T^{2} \)
29 \( 1 + 1.96T + 29T^{2} \)
31 \( 1 + 4.66T + 31T^{2} \)
37 \( 1 + 3.44iT - 37T^{2} \)
41 \( 1 + 2.68T + 41T^{2} \)
43 \( 1 - 10.8iT - 43T^{2} \)
47 \( 1 - 9.67iT - 47T^{2} \)
53 \( 1 - 5.97iT - 53T^{2} \)
59 \( 1 - 9.18T + 59T^{2} \)
61 \( 1 + 5.69T + 61T^{2} \)
67 \( 1 - 11.8iT - 67T^{2} \)
71 \( 1 + 0.530T + 71T^{2} \)
73 \( 1 - 8.20iT - 73T^{2} \)
79 \( 1 - 9.12T + 79T^{2} \)
83 \( 1 - 2.96iT - 83T^{2} \)
89 \( 1 + 10.8T + 89T^{2} \)
97 \( 1 - 13.5iT - 97T^{2} \)
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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

−9.369930275893107590212754590231, −8.785494652539166358894589560106, −7.65380167774833575482924637569, −7.15399756172304116241798716909, −6.42477786185692110247332542839, −5.09286995791423445432074738203, −4.37479459828275619236519791774, −3.80960771388544103252943880203, −2.69250627110184236685406914866, −1.22934203755139800834297408607, 0.57315200086914484529003403986, 1.71091424524070886531450496382, 3.30750323776740445359314724741, 3.72968569403430978022791665600, 5.04957444281548364388639488085, 5.63379165866011215588285733182, 6.88107077347425246594777321878, 7.52845328202880899677465673413, 7.936389784296763281983044070092, 8.811880893387728463474213466934

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