Properties

Label 2-1960-280.219-c0-0-0
Degree $2$
Conductor $1960$
Sign $0.605 - 0.795i$
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  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (−0.5 + 0.866i)5-s − 0.999·8-s + (−0.5 + 0.866i)9-s + (0.499 + 0.866i)10-s + (0.5 + 0.866i)11-s − 13-s + (−0.5 + 0.866i)16-s + (0.499 + 0.866i)18-s + (−0.5 + 0.866i)19-s + 0.999·20-s + 0.999·22-s + (−0.5 + 0.866i)23-s + (−0.499 − 0.866i)25-s + (−0.5 + 0.866i)26-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (−0.5 + 0.866i)5-s − 0.999·8-s + (−0.5 + 0.866i)9-s + (0.499 + 0.866i)10-s + (0.5 + 0.866i)11-s − 13-s + (−0.5 + 0.866i)16-s + (0.499 + 0.866i)18-s + (−0.5 + 0.866i)19-s + 0.999·20-s + 0.999·22-s + (−0.5 + 0.866i)23-s + (−0.499 − 0.866i)25-s + (−0.5 + 0.866i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1960\)    =    \(2^{3} \cdot 5 \cdot 7^{2}\)
Sign: $0.605 - 0.795i$
Analytic conductor: \(0.978167\)
Root analytic conductor: \(0.989023\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1960} (1059, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1960,\ (\ :0),\ 0.605 - 0.795i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8430574359\)
\(L(\frac12)\) \(\approx\) \(0.8430574359\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 \)
good3 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + T + T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 - T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
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 \( 1 - T^{2} \)
89 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
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

−9.903016701238506036072904180450, −8.825314246377564816052352640215, −7.87013434859997303068917552513, −7.14933828685915463819293228801, −6.19170417659035679578922542186, −5.29941768361267823946218681733, −4.41936262204184481569765927544, −3.65057539839967050742999293568, −2.60555079813967805201716209879, −1.87546853892448540342866567915, 0.49895080111073473926780434137, 2.64981295554159799151061999250, 3.72097064814655019970564345140, 4.40631370173048945845939277510, 5.26641287380979921991738051875, 6.07864227645811768284050192545, 6.79473770906631067789140861898, 7.70304075584915321926070795753, 8.447999485081896129564081944490, 9.040653047713604678570768425618

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