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Properties

Label 2-1960-7.4-c1-0-14
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  + (−1.5 + 2.59i)3-s + (0.5 + 0.866i)5-s + (−3 − 5.19i)9-s + (2.5 − 4.33i)11-s + 5·13-s − 3·15-s + (−3.5 + 6.06i)17-s + (−1 − 1.73i)19-s + (1 + 1.73i)23-s + (−0.499 + 0.866i)25-s + 9·27-s + 7·29-s + (2 − 3.46i)31-s + (7.50 + 12.9i)33-s + (3 + 5.19i)37-s + ⋯
L(s)  = 1  + (−0.866 + 1.49i)3-s + (0.223 + 0.387i)5-s + (−1 − 1.73i)9-s + (0.753 − 1.30i)11-s + 1.38·13-s − 0.774·15-s + (−0.848 + 1.47i)17-s + (−0.229 − 0.397i)19-s + (0.208 + 0.361i)23-s + (−0.0999 + 0.173i)25-s + 1.73·27-s + 1.29·29-s + (0.359 − 0.622i)31-s + (1.30 + 2.26i)33-s + (0.493 + 0.854i)37-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: $\chi_{1960} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1960,\ (\ :1/2),\ -0.266 - 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.398910303\)
\(L(\frac12)\) \(\approx\) \(1.398910303\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.5 - 2.59i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (-2.5 + 4.33i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 5T + 13T^{2} \)
17 \( 1 + (3.5 - 6.06i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1 + 1.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1 - 1.73i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 7T + 29T^{2} \)
31 \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3 - 5.19i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 12T + 41T^{2} \)
43 \( 1 + 2T + 43T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2 - 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2 - 3.46i)T + (-30.5 + 52.8i)T^{2} \)
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 \( 1 + (-1.5 - 2.59i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 4T + 83T^{2} \)
89 \( 1 + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 13T + 97T^{2} \)
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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

−9.419405070665591639616921151943, −8.820918428669247315650459461074, −8.179772671739186686735537662119, −6.39723349026311220489765572830, −6.29364345372281787405357620537, −5.51779709043169992480590680719, −4.29098332677716516133742311035, −3.89863924625019302538537302894, −2.92305349634088367619143291018, −1.04527688214650117404540081771, 0.74147257859353841690614118403, 1.62458168181415894340280030754, 2.58696421747571824449021480651, 4.24201291488653967627097366766, 5.02125871082504210783273321456, 6.03776595078994905001827323637, 6.60295424472382285492880394567, 7.15523333425596544443829056455, 8.023789359013322435640585830623, 8.859469700794821283040768940734

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