Properties

Label 2-960-12.11-c1-0-14
Degree $2$
Conductor $960$
Sign $0.962 + 0.270i$
Analytic cond. $7.66563$
Root an. cond. $2.76868$
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  + (−0.468 + 1.66i)3-s i·5-s − 0.936i·7-s + (−2.56 − 1.56i)9-s + 4.27·11-s − 3.12·13-s + (1.66 + 0.468i)15-s − 2i·17-s − 4.27i·19-s + (1.56 + 0.438i)21-s + 7.60·23-s − 25-s + (3.80 − 3.54i)27-s − 5.12i·29-s + 2.39i·31-s + ⋯
L(s)  = 1  + (−0.270 + 0.962i)3-s − 0.447i·5-s − 0.353i·7-s + (−0.853 − 0.520i)9-s + 1.28·11-s − 0.866·13-s + (0.430 + 0.120i)15-s − 0.485i·17-s − 0.979i·19-s + (0.340 + 0.0956i)21-s + 1.58·23-s − 0.200·25-s + (0.731 − 0.681i)27-s − 0.951i·29-s + 0.430i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 + 0.270i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.962 + 0.270i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $0.962 + 0.270i$
Analytic conductor: \(7.66563\)
Root analytic conductor: \(2.76868\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :1/2),\ 0.962 + 0.270i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.36921 - 0.188575i\)
\(L(\frac12)\) \(\approx\) \(1.36921 - 0.188575i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.468 - 1.66i)T \)
5 \( 1 + iT \)
good7 \( 1 + 0.936iT - 7T^{2} \)
11 \( 1 - 4.27T + 11T^{2} \)
13 \( 1 + 3.12T + 13T^{2} \)
17 \( 1 + 2iT - 17T^{2} \)
19 \( 1 + 4.27iT - 19T^{2} \)
23 \( 1 - 7.60T + 23T^{2} \)
29 \( 1 + 5.12iT - 29T^{2} \)
31 \( 1 - 2.39iT - 31T^{2} \)
37 \( 1 - 3.12T + 37T^{2} \)
41 \( 1 + 7.12iT - 41T^{2} \)
43 \( 1 - 1.46iT - 43T^{2} \)
47 \( 1 + 0.936T + 47T^{2} \)
53 \( 1 - 4.24iT - 53T^{2} \)
59 \( 1 - 7.19T + 59T^{2} \)
61 \( 1 - 5.12T + 61T^{2} \)
67 \( 1 - 5.20iT - 67T^{2} \)
71 \( 1 - 6.67T + 71T^{2} \)
73 \( 1 + 8.24T + 73T^{2} \)
79 \( 1 + 9.06iT - 79T^{2} \)
83 \( 1 - 4.68T + 83T^{2} \)
89 \( 1 - 6.24iT - 89T^{2} \)
97 \( 1 + 6T + 97T^{2} \)
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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

−9.858160910804712880237071604244, −9.217963667427776956807438553996, −8.702855138270988188760249646691, −7.34444037504840353149720687679, −6.58748816239931785090215345498, −5.41656359745453817954567637505, −4.66308033698889187253892795526, −3.90350038298692946043145910251, −2.69135329919488648704785523185, −0.77590398662705480563984510010, 1.26488667624868477988341166564, 2.44634767420328228480978451930, 3.59895468519661589364563265603, 4.97348350548647189558685672771, 5.95885326628215205433745262776, 6.72073663608115930487984208005, 7.35227506024364395330850206479, 8.332226316647627495100746433617, 9.139725326501899269093799892854, 10.07732146022311619675091693458

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