Properties

Label 2-1960-7.2-c1-0-32
Degree $2$
Conductor $1960$
Sign $-0.991 - 0.126i$
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  + (−1.18 − 2.05i)3-s + (0.5 − 0.866i)5-s + (−1.31 + 2.27i)9-s + (−3.18 − 5.51i)11-s + 4.37·13-s − 2.37·15-s + (0.186 + 0.322i)17-s + (2.37 − 4.10i)19-s + (2.37 − 4.10i)23-s + (−0.499 − 0.866i)25-s − 0.883·27-s − 4.37·29-s + (4 + 6.92i)31-s + (−7.55 + 13.0i)33-s + (1 − 1.73i)37-s + ⋯
L(s)  = 1  + (−0.684 − 1.18i)3-s + (0.223 − 0.387i)5-s + (−0.437 + 0.758i)9-s + (−0.960 − 1.66i)11-s + 1.21·13-s − 0.612·15-s + (0.0451 + 0.0781i)17-s + (0.544 − 0.942i)19-s + (0.494 − 0.856i)23-s + (−0.0999 − 0.173i)25-s − 0.169·27-s − 0.811·29-s + (0.718 + 1.24i)31-s + (−1.31 + 2.27i)33-s + (0.164 − 0.284i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1960\)    =    \(2^{3} \cdot 5 \cdot 7^{2}\)
Sign: $-0.991 - 0.126i$
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: \((2,\ 1960,\ (\ :1/2),\ -0.991 - 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.087716660\)
\(L(\frac12)\) \(\approx\) \(1.087716660\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.18 + 2.05i)T + (-1.5 + 2.59i)T^{2} \)
11 \( 1 + (3.18 + 5.51i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 4.37T + 13T^{2} \)
17 \( 1 + (-0.186 - 0.322i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.37 + 4.10i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.37 + 4.10i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 4.37T + 29T^{2} \)
31 \( 1 + (-4 - 6.92i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1 + 1.73i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 6.74T + 41T^{2} \)
43 \( 1 + 8.74T + 43T^{2} \)
47 \( 1 + (-3.55 + 6.16i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.37 + 9.30i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4 + 6.92i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.37 + 2.37i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2 - 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + (-3 - 5.19i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (7.55 - 13.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 9.48T + 83T^{2} \)
89 \( 1 + (7.37 - 12.7i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 9.86T + 97T^{2} \)
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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

−8.406107741349939228258572057317, −8.207522200157620078191060226411, −6.99387486087086616546683296157, −6.43399119389322619070974531739, −5.62490633333546792270366503254, −5.11558015129126344475580157147, −3.63270243102510229865068218828, −2.62585239714198096983181825339, −1.27071357407278623969741918551, −0.47005717657123338054296365084, 1.65599864528202014112322764702, 2.98464112768167879233934969126, 4.02303340418776986821905333479, 4.68558004244876747260462772387, 5.59270882168321379392135761614, 6.10169129274440323748266669649, 7.34798514186594101591497838276, 7.86973464859993603093098510797, 9.155668419870614759599823208259, 9.809405898956068501284283894464

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