Properties

Label 2-1960-5.4-c1-0-54
Degree $2$
Conductor $1960$
Sign $0.621 + 0.783i$
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  + 0.363i·3-s + (1.75 − 1.38i)5-s + 2.86·9-s + 5.14·11-s − 4.64i·13-s + (0.504 + 0.636i)15-s − 3.86i·17-s − 0.778·19-s − 5.00i·23-s + (1.14 − 4.86i)25-s + 2.13i·27-s − 9.42·29-s − 4.72·31-s + 1.86i·33-s + 6i·37-s + ⋯
L(s)  = 1  + 0.209i·3-s + (0.783 − 0.621i)5-s + 0.955·9-s + 1.55·11-s − 1.28i·13-s + (0.130 + 0.164i)15-s − 0.938i·17-s − 0.178·19-s − 1.04i·23-s + (0.228 − 0.973i)25-s + 0.410i·27-s − 1.74·29-s − 0.848·31-s + 0.325i·33-s + 0.986i·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.621 + 0.783i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.621 + 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1960\)    =    \(2^{3} \cdot 5 \cdot 7^{2}\)
Sign: $0.621 + 0.783i$
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: \((2,\ 1960,\ (\ :1/2),\ 0.621 + 0.783i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.323730493\)
\(L(\frac12)\) \(\approx\) \(2.323730493\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.75 + 1.38i)T \)
7 \( 1 \)
good3 \( 1 - 0.363iT - 3T^{2} \)
11 \( 1 - 5.14T + 11T^{2} \)
13 \( 1 + 4.64iT - 13T^{2} \)
17 \( 1 + 3.86iT - 17T^{2} \)
19 \( 1 + 0.778T + 19T^{2} \)
23 \( 1 + 5.00iT - 23T^{2} \)
29 \( 1 + 9.42T + 29T^{2} \)
31 \( 1 + 4.72T + 31T^{2} \)
37 \( 1 - 6iT - 37T^{2} \)
41 \( 1 - 1.00T + 41T^{2} \)
43 \( 1 - 7.00iT - 43T^{2} \)
47 \( 1 - 11.4iT - 47T^{2} \)
53 \( 1 + 7.55iT - 53T^{2} \)
59 \( 1 - 12.5T + 59T^{2} \)
61 \( 1 + 11.5T + 61T^{2} \)
67 \( 1 + 11.7iT - 67T^{2} \)
71 \( 1 - 2.72T + 71T^{2} \)
73 \( 1 - 5.00iT - 73T^{2} \)
79 \( 1 + 5.68T + 79T^{2} \)
83 \( 1 + 4.67iT - 83T^{2} \)
89 \( 1 + 2.82T + 89T^{2} \)
97 \( 1 - 1.58iT - 97T^{2} \)
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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

−9.383065844615053653535905272249, −8.407253137388055335411431435880, −7.48546629499239363737281836078, −6.62492877730884455781796808760, −5.88213854718146513437461114852, −4.95990460585336421976045504228, −4.25514712357181279959073905360, −3.21504636662760560836523751796, −1.88794548920622952583164233758, −0.900374415020395066097176543167, 1.59840149186568241818746655758, 1.96195327317938833202945484473, 3.74079023681433928662615165964, 4.04186850934380039037985150141, 5.49923469220541870913622056156, 6.20685870337020542290426291953, 7.06373905899661990331066833702, 7.29770565930252871153388275466, 8.806371720575594838669938103758, 9.294538376861053956113927240591

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