Properties

Label 2-1960-5.3-c0-0-0
Degree $2$
Conductor $1960$
Sign $0.525 - 0.850i$
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)3-s + (0.707 + 0.707i)5-s + 11-s + (−0.707 − 0.707i)13-s + 1.00i·15-s + (−0.707 + 0.707i)17-s + 1.41i·19-s + 1.00i·25-s + (0.707 − 0.707i)27-s i·29-s + 1.41·31-s + (0.707 + 0.707i)33-s − 1.00i·39-s − 1.41·41-s + (−1 − i)43-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)3-s + (0.707 + 0.707i)5-s + 11-s + (−0.707 − 0.707i)13-s + 1.00i·15-s + (−0.707 + 0.707i)17-s + 1.41i·19-s + 1.00i·25-s + (0.707 − 0.707i)27-s i·29-s + 1.41·31-s + (0.707 + 0.707i)33-s − 1.00i·39-s − 1.41·41-s + (−1 − i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1960\)    =    \(2^{3} \cdot 5 \cdot 7^{2}\)
Sign: $0.525 - 0.850i$
Analytic conductor: \(0.978167\)
Root analytic conductor: \(0.989023\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1960} (393, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1960,\ (\ :0),\ 0.525 - 0.850i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.661219452\)
\(L(\frac12)\) \(\approx\) \(1.661219452\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−9.808676771574831404325543310557, −8.700757147792071855470255198362, −8.213996034213438079098300838178, −7.03852379862538316210765662644, −6.34978255089868026487900151360, −5.59075646136150993324910360243, −4.37322042557541065139711921262, −3.63696915767250977084305921486, −2.80856031165082129564887514783, −1.74746769945061205058032668569, 1.30331451197242421052466012012, 2.20330101542419209099883202251, 3.06473609289237927131530240221, 4.67065651444874476297985607706, 4.88833739359896774435953959533, 6.43128032781403388384822115981, 6.78455456199932045878558263786, 7.70939243015627270064818668767, 8.642214106721314138006257428671, 9.130574713424885829263015637640

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