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Properties

Label 2-1960-7.2-c1-0-17
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: $\chi_{1960} (961, \cdot )$
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

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

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