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Properties

Label 2-1960-7.2-c1-0-37
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  + (−1.43 − 2.49i)3-s + (−0.5 + 0.866i)5-s + (−2.64 + 4.58i)9-s + (−0.732 − 1.26i)11-s + 2.22·13-s + 2.87·15-s + (−3.63 − 6.29i)17-s + (2.74 − 4.75i)19-s + (−1.25 + 2.17i)23-s + (−0.499 − 0.866i)25-s + 6.60·27-s − 7.12·29-s + (3.25 + 5.64i)31-s + (−2.11 + 3.65i)33-s + (−3.45 + 5.97i)37-s + ⋯
L(s)  = 1  + (−0.831 − 1.43i)3-s + (−0.223 + 0.387i)5-s + (−0.882 + 1.52i)9-s + (−0.220 − 0.382i)11-s + 0.616·13-s + 0.743·15-s + (−0.881 − 1.52i)17-s + (0.629 − 1.09i)19-s + (−0.262 + 0.454i)23-s + (−0.0999 − 0.173i)25-s + 1.27·27-s − 1.32·29-s + (0.585 + 1.01i)31-s + (−0.367 + 0.636i)33-s + (−0.567 + 0.982i)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: $\chi_{1960} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1960,\ (\ :1/2),\ -0.198 - 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.02825255743\)
\(L(\frac12)\) \(\approx\) \(0.02825255743\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.43 + 2.49i)T + (-1.5 + 2.59i)T^{2} \)
11 \( 1 + (0.732 + 1.26i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2.22T + 13T^{2} \)
17 \( 1 + (3.63 + 6.29i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.74 + 4.75i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.25 - 2.17i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 7.12T + 29T^{2} \)
31 \( 1 + (-3.25 - 5.64i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.45 - 5.97i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 11.3T + 41T^{2} \)
43 \( 1 - 3.31T + 43T^{2} \)
47 \( 1 + (-4.18 + 7.24i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.50 - 6.06i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.53 + 7.85i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.60 - 9.71i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.20 - 5.55i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 10.5T + 71T^{2} \)
73 \( 1 + (5.26 + 9.11i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.09 - 8.82i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 16.4T + 83T^{2} \)
89 \( 1 + (4.91 - 8.51i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 2.09T + 97T^{2} \)
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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

−8.494845222912286018267389134682, −7.48446426475257251522927852012, −7.05839563124330324417155347026, −6.44030742698590874024969166464, −5.52515023224277075831265765072, −4.83473537417920828009620863593, −3.34511330352312419376516854395, −2.38456998118199953554666967731, −1.17727676950263692883848808400, −0.01237775012716109658821179961, 1.79434251417871329662274316680, 3.52348415943636163870067097216, 4.07747738907532883745323261500, 4.78273971396552565125896266567, 5.79474272422876292384745551241, 6.12264225667382792064440199339, 7.45479860296315407125191379818, 8.404800569948532617881018320460, 9.064381511666648693173702648786, 9.891696757143250687313188563169

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