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Properties

Label 2-960-15.14-c2-0-63
Degree $2$
Conductor $960$
Sign $0.262 + 0.964i$
Analytic cond. $26.1581$
Root an. cond. $5.11449$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.23 + 2i)3-s + (2.23 − 4.47i)5-s − 8i·7-s + (1.00 − 8.94i)9-s + 8.94i·11-s + 12i·13-s + (3.94 + 14.4i)15-s + 31.3·17-s − 6·19-s + (16 + 17.8i)21-s + 4.47·23-s + (−15.0 − 20.0i)25-s + (15.6 + 22.0i)27-s − 26.8i·29-s − 34·31-s + ⋯
L(s)  = 1  + (−0.745 + 0.666i)3-s + (0.447 − 0.894i)5-s − 1.14i·7-s + (0.111 − 0.993i)9-s + 0.813i·11-s + 0.923i·13-s + (0.262 + 0.964i)15-s + 1.84·17-s − 0.315·19-s + (0.761 + 0.851i)21-s + 0.194·23-s + (−0.600 − 0.800i)25-s + (0.579 + 0.814i)27-s − 0.925i·29-s − 1.09·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.262 + 0.964i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.262 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $0.262 + 0.964i$
Analytic conductor: \(26.1581\)
Root analytic conductor: \(5.11449\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :1),\ 0.262 + 0.964i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.360836266\)
\(L(\frac12)\) \(\approx\) \(1.360836266\)
\(L(2)\) 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 \( 1 + (2.23 - 2i)T \)
5 \( 1 + (-2.23 + 4.47i)T \)
good7 \( 1 + 8iT - 49T^{2} \)
11 \( 1 - 8.94iT - 121T^{2} \)
13 \( 1 - 12iT - 169T^{2} \)
17 \( 1 - 31.3T + 289T^{2} \)
19 \( 1 + 6T + 361T^{2} \)
23 \( 1 - 4.47T + 529T^{2} \)
29 \( 1 + 26.8iT - 841T^{2} \)
31 \( 1 + 34T + 961T^{2} \)
37 \( 1 + 44iT - 1.36e3T^{2} \)
41 \( 1 + 17.8iT - 1.68e3T^{2} \)
43 \( 1 - 28iT - 1.84e3T^{2} \)
47 \( 1 - 4.47T + 2.20e3T^{2} \)
53 \( 1 - 40.2T + 2.80e3T^{2} \)
59 \( 1 + 98.3iT - 3.48e3T^{2} \)
61 \( 1 + 74T + 3.72e3T^{2} \)
67 \( 1 + 92iT - 4.48e3T^{2} \)
71 \( 1 - 53.6iT - 5.04e3T^{2} \)
73 \( 1 + 56iT - 5.32e3T^{2} \)
79 \( 1 - 78T + 6.24e3T^{2} \)
83 \( 1 - 102.T + 6.88e3T^{2} \)
89 \( 1 - 17.8iT - 7.92e3T^{2} \)
97 \( 1 + 32iT - 9.40e3T^{2} \)
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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

−9.654718963501792890711051385215, −9.234010964513591537392382025872, −7.905830503584137276598437586927, −7.07572054146708475725860418196, −6.05642587410979766359741297600, −5.19307022530198434198306701006, −4.38050865857605679457396053425, −3.68376029292752507421317710368, −1.71328367436028077459072079993, −0.53798223621897566175362019500, 1.21823706284386630488458937993, 2.55121825752279001289322625441, 3.37059281104143024782426337900, 5.37000575542671209361176081780, 5.60997791787082784396805071162, 6.43913582198432812563487540614, 7.42632482580924554390145954382, 8.170848515485310272483211141984, 9.172770606030593009933456908218, 10.32645392541165564422111335429

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