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Properties

Label 2-1960-280.277-c0-0-1
Degree $2$
Conductor $1960$
Sign $0.477 - 0.878i$
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  + (−0.258 + 0.965i)2-s + (1.78 − 0.478i)3-s + (−0.866 − 0.499i)4-s + (0.130 + 0.991i)5-s + 1.84i·6-s + (0.707 − 0.707i)8-s + (2.09 − 1.20i)9-s + (−0.991 − 0.130i)10-s + (−1.78 − 0.478i)12-s + (0.541 + 0.541i)13-s + (0.707 + 1.70i)15-s + (0.500 + 0.866i)16-s + (0.624 + 2.33i)18-s + (−0.382 − 0.662i)19-s + (0.382 − 0.923i)20-s + ⋯
L(s)  = 1  + (−0.258 + 0.965i)2-s + (1.78 − 0.478i)3-s + (−0.866 − 0.499i)4-s + (0.130 + 0.991i)5-s + 1.84i·6-s + (0.707 − 0.707i)8-s + (2.09 − 1.20i)9-s + (−0.991 − 0.130i)10-s + (−1.78 − 0.478i)12-s + (0.541 + 0.541i)13-s + (0.707 + 1.70i)15-s + (0.500 + 0.866i)16-s + (0.624 + 2.33i)18-s + (−0.382 − 0.662i)19-s + (0.382 − 0.923i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.477 - 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.477 - 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1960\)    =    \(2^{3} \cdot 5 \cdot 7^{2}\)
Sign: $0.477 - 0.878i$
Analytic conductor: \(0.978167\)
Root analytic conductor: \(0.989023\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1960} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1960,\ (\ :0),\ 0.477 - 0.878i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.773496033\)
\(L(\frac12)\) \(\approx\) \(1.773496033\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 - 0.965i)T \)
5 \( 1 + (-0.130 - 0.991i)T \)
7 \( 1 \)
good3 \( 1 + (-1.78 + 0.478i)T + (0.866 - 0.5i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.541 - 0.541i)T + iT^{2} \)
17 \( 1 + (0.866 - 0.5i)T^{2} \)
19 \( 1 + (0.382 + 0.662i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.866 + 0.5i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (-0.866 - 0.5i)T^{2} \)
53 \( 1 + (0.866 - 0.5i)T^{2} \)
59 \( 1 + (0.923 - 1.60i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (-1.60 + 0.923i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.866 + 0.5i)T^{2} \)
71 \( 1 + 1.41T + T^{2} \)
73 \( 1 + (-0.866 + 0.5i)T^{2} \)
79 \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.541 + 0.541i)T + iT^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
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

−9.206502541479254770046374784743, −8.586569153926780861722400014747, −7.940895561272564378668553742959, −7.23866345210558587196270195883, −6.66918039860157906895906794209, −5.93835261326952378869919978467, −4.37182133645491597151542511672, −3.71181468502925727960372083469, −2.65264540091493611685677136245, −1.67503005581090212376971684606, 1.48651922830838490108536160512, 2.27096948595198389222278382659, 3.36922010947184365528067311947, 3.98522762492858732543102282085, 4.69885215425056315451816280952, 5.80279658442405519467015771373, 7.50271263465306409791046276844, 8.155576707214416255234884589211, 8.562573052744608074947307294162, 9.194874357922720108277571100010

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