Properties

Label 2-1960-7.4-c1-0-32
Degree $2$
Conductor $1960$
Sign $-0.198 + 0.980i$
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  + (−1.20 + 2.09i)3-s + (0.5 + 0.866i)5-s + (−1.41 − 2.44i)9-s + (0.5 − 0.866i)11-s − 0.414·13-s − 2.41·15-s + (−1.20 + 2.09i)17-s + (−1 − 1.73i)19-s + (−3.12 − 5.40i)23-s + (−0.499 + 0.866i)25-s − 0.414·27-s + 29-s + (−5.12 + 8.87i)31-s + (1.20 + 2.09i)33-s + (−5.94 − 10.3i)37-s + ⋯
L(s)  = 1  + (−0.696 + 1.20i)3-s + (0.223 + 0.387i)5-s + (−0.471 − 0.816i)9-s + (0.150 − 0.261i)11-s − 0.114·13-s − 0.623·15-s + (−0.292 + 0.507i)17-s + (−0.229 − 0.397i)19-s + (−0.650 − 1.12i)23-s + (−0.0999 + 0.173i)25-s − 0.0797·27-s + 0.185·29-s + (−0.919 + 1.59i)31-s + (0.210 + 0.363i)33-s + (−0.978 − 1.69i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.198 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.198 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1960\)    =    \(2^{3} \cdot 5 \cdot 7^{2}\)
Sign: $-0.198 + 0.980i$
Analytic conductor: \(15.6506\)
Root analytic conductor: \(3.95609\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1960} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1960,\ (\ :1/2),\ -0.198 + 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.03882525152\)
\(L(\frac12)\) \(\approx\) \(0.03882525152\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.20 - 2.09i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 0.414T + 13T^{2} \)
17 \( 1 + (1.20 - 2.09i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1 + 1.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.12 + 5.40i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - T + 29T^{2} \)
31 \( 1 + (5.12 - 8.87i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.94 + 10.3i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 4.58T + 41T^{2} \)
43 \( 1 + 11.6T + 43T^{2} \)
47 \( 1 + (3.79 + 6.56i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.29 - 5.70i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.878 - 1.52i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.41 - 5.91i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.707 + 1.22i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 2.48T + 71T^{2} \)
73 \( 1 + (-5.41 + 9.37i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.67 - 2.89i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 11.3T + 83T^{2} \)
89 \( 1 + (4.82 + 8.36i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 14.0T + 97T^{2} \)
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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

−9.041598818406488925233335792371, −8.449910923433796645479309620313, −7.21743507195144389196716702340, −6.44497068591366004737415805289, −5.64763767809327147424584274986, −4.90643311536990496119454909788, −4.08485258834485946496757842160, −3.26983867463519943360798286105, −1.95598400487570619758314104402, −0.01514302682623437574153683483, 1.36494022191740820405549841537, 2.13119956313895907924894190081, 3.55972982862249453686778384503, 4.74472263817793747892862102571, 5.57010975786192793271425688755, 6.29398014168821624788690407880, 6.97097820976062022044695147402, 7.74967386511403776232843526125, 8.392298909377652715306775368734, 9.531752987327477292014290142436

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