Properties

Label 2-1960-40.37-c0-0-2
Degree $2$
Conductor $1960$
Sign $0.584 - 0.811i$
Analytic cond. $0.978167$
Root an. cond. $0.989023$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 − 0.707i)2-s + (1.30 + 1.30i)3-s + 1.00i·4-s + (0.923 + 0.382i)5-s − 1.84i·6-s + (0.707 − 0.707i)8-s + 2.41i·9-s + (−0.382 − 0.923i)10-s + (−1.30 + 1.30i)12-s + (−0.541 − 0.541i)13-s + (0.707 + 1.70i)15-s − 1.00·16-s + (1.70 − 1.70i)18-s − 0.765·19-s + (−0.382 + 0.923i)20-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s + (1.30 + 1.30i)3-s + 1.00i·4-s + (0.923 + 0.382i)5-s − 1.84i·6-s + (0.707 − 0.707i)8-s + 2.41i·9-s + (−0.382 − 0.923i)10-s + (−1.30 + 1.30i)12-s + (−0.541 − 0.541i)13-s + (0.707 + 1.70i)15-s − 1.00·16-s + (1.70 − 1.70i)18-s − 0.765·19-s + (−0.382 + 0.923i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.584 - 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.584 - 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1960\)    =    \(2^{3} \cdot 5 \cdot 7^{2}\)
Sign: $0.584 - 0.811i$
Analytic conductor: \(0.978167\)
Root analytic conductor: \(0.989023\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1960} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1960,\ (\ :0),\ 0.584 - 0.811i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.428224617\)
\(L(\frac12)\) \(\approx\) \(1.428224617\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 + (-0.923 - 0.382i)T \)
7 \( 1 \)
good3 \( 1 + (-1.30 - 1.30i)T + iT^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + (0.541 + 0.541i)T + iT^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + 0.765T + T^{2} \)
23 \( 1 + (-1 + i)T - iT^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + 1.84T + T^{2} \)
61 \( 1 + 1.84iT - T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 + 1.41T + T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + 1.41iT - T^{2} \)
83 \( 1 + (-0.541 - 0.541i)T + iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + iT^{2} \)
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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

−9.339307722413369425934876298506, −9.099419973380599849809065007881, −8.262262278158031216621242948399, −7.55373263640289557081333430545, −6.49321663201466847667464683112, −5.08372625184832739370626747581, −4.38185669204567361948563505248, −3.27361357961696287485301442741, −2.73822029929784028844690006331, −1.91549915719754936815807246213, 1.26036351910011343751498745850, 1.99139846824909120045518665821, 2.89929859717625044337624414899, 4.45984546038362720506879316248, 5.62752648938212831169448802090, 6.41393128536192617747070708613, 7.08599176562412332943108379978, 7.65795768745808642153017316493, 8.555612032339964593949407240147, 9.030474709024362152585500078749

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