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Properties

Label 2-1960-7.4-c1-0-31
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  + (1.28 − 2.21i)3-s + (0.5 + 0.866i)5-s + (−1.78 − 3.08i)9-s + (1.28 − 2.21i)11-s + 5.68·13-s + 2.56·15-s + (1.71 − 2.97i)17-s + (0.561 + 0.972i)19-s + (2.56 + 4.43i)23-s + (−0.499 + 0.866i)25-s − 1.43·27-s + 4.56·29-s + (−5.12 + 8.87i)31-s + (−3.28 − 5.68i)33-s + (−4.12 − 7.14i)37-s + ⋯
L(s)  = 1  + (0.739 − 1.28i)3-s + (0.223 + 0.387i)5-s + (−0.593 − 1.02i)9-s + (0.386 − 0.668i)11-s + 1.57·13-s + 0.661·15-s + (0.416 − 0.722i)17-s + (0.128 + 0.223i)19-s + (0.534 + 0.925i)23-s + (−0.0999 + 0.173i)25-s − 0.276·27-s + 0.847·29-s + (−0.920 + 1.59i)31-s + (−0.571 − 0.989i)33-s + (−0.677 − 1.17i)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

\(L(1)\) \(\approx\) \(2.671598237\)
\(L(\frac12)\) \(\approx\) \(2.671598237\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.28 + 2.21i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (-1.28 + 2.21i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 5.68T + 13T^{2} \)
17 \( 1 + (-1.71 + 2.97i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.561 - 0.972i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.56 - 4.43i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 4.56T + 29T^{2} \)
31 \( 1 + (5.12 - 8.87i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.12 + 7.14i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 7.12T + 41T^{2} \)
43 \( 1 - 1.12T + 43T^{2} \)
47 \( 1 + (-3.28 - 5.68i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.43 + 4.22i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2 - 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.56 + 13.0i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.12 + 12.3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (-6.12 + 10.6i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.84 - 10.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 + (1.56 + 2.70i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 13.6T + 97T^{2} \)
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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

−8.810702205361890568373761915447, −8.264864696267023615701986004034, −7.40424360090084722162621300087, −6.77329055118000203873575899924, −6.09495180027569976670453402781, −5.18165680818807270454175604096, −3.52421233915913398641555951370, −3.16831753622017040497641810319, −1.83930142815929609664240333550, −1.05184547559119673258041889096, 1.35491559699654480310465303788, 2.67882262194379045017227871740, 3.74362940032869252125142011956, 4.21266694781622261577274645182, 5.13472693775090626074507189276, 6.05294578203701796113130799455, 6.96978054196769540211173168946, 8.240214387592173005050090330598, 8.642262541205430372581140792998, 9.275006084432700848919355967668

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