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Properties

Label 2-960-120.29-c2-0-43
Degree $2$
Conductor $960$
Sign $0.958 + 0.285i$
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  + (−2.90 + 0.751i)3-s + (1.38 − 4.80i)5-s + 3.03i·7-s + (7.86 − 4.36i)9-s + 11.4·11-s + 16.9·13-s + (−0.413 + 14.9i)15-s − 18.1·17-s + 16.9i·19-s + (−2.28 − 8.82i)21-s − 22.3·23-s + (−21.1 − 13.3i)25-s + (−19.5 + 18.5i)27-s + 17.8·29-s + 16.9·31-s + ⋯
L(s)  = 1  + (−0.968 + 0.250i)3-s + (0.277 − 0.960i)5-s + 0.434i·7-s + (0.874 − 0.485i)9-s + 1.04·11-s + 1.30·13-s + (−0.0275 + 0.999i)15-s − 1.06·17-s + 0.891i·19-s + (−0.108 − 0.420i)21-s − 0.971·23-s + (−0.846 − 0.532i)25-s + (−0.724 + 0.688i)27-s + 0.617·29-s + 0.547·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 + 0.285i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.958 + 0.285i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $0.958 + 0.285i$
Analytic conductor: \(26.1581\)
Root analytic conductor: \(5.11449\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (929, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :1),\ 0.958 + 0.285i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.550792746\)
\(L(\frac12)\) \(\approx\) \(1.550792746\)
\(L(2)\) 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 \( 1 + (2.90 - 0.751i)T \)
5 \( 1 + (-1.38 + 4.80i)T \)
good7 \( 1 - 3.03iT - 49T^{2} \)
11 \( 1 - 11.4T + 121T^{2} \)
13 \( 1 - 16.9T + 169T^{2} \)
17 \( 1 + 18.1T + 289T^{2} \)
19 \( 1 - 16.9iT - 361T^{2} \)
23 \( 1 + 22.3T + 529T^{2} \)
29 \( 1 - 17.8T + 841T^{2} \)
31 \( 1 - 16.9T + 961T^{2} \)
37 \( 1 - 1.21T + 1.36e3T^{2} \)
41 \( 1 + 42.0iT - 1.68e3T^{2} \)
43 \( 1 + 1.98T + 1.84e3T^{2} \)
47 \( 1 - 52.4T + 2.20e3T^{2} \)
53 \( 1 + 8.58iT - 2.80e3T^{2} \)
59 \( 1 - 93.0T + 3.48e3T^{2} \)
61 \( 1 - 80.4iT - 3.72e3T^{2} \)
67 \( 1 - 38.1T + 4.48e3T^{2} \)
71 \( 1 - 109. iT - 5.04e3T^{2} \)
73 \( 1 + 28.2iT - 5.32e3T^{2} \)
79 \( 1 + 56.1T + 6.24e3T^{2} \)
83 \( 1 + 126. iT - 6.88e3T^{2} \)
89 \( 1 + 126. iT - 7.92e3T^{2} \)
97 \( 1 - 153. iT - 9.40e3T^{2} \)
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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

−9.851617955692312727464133272623, −8.876497945032059136905421888974, −8.474623028689952880636641335039, −7.01830918618479246246681155451, −6.03227161402850996665767107934, −5.70604498740672969935968996839, −4.40936821346881925563611981586, −3.88449521867354737056010607634, −1.87751731885665154818640020444, −0.812420258551469753959042262622, 0.900421048017397224962619958686, 2.18685495529277093208215155560, 3.69659495870248589592863447888, 4.51072826050826784299369157969, 5.84972025055235669749071806140, 6.54666399877807336430246186270, 6.91460351180637273445298678381, 8.080763520355169635634959590444, 9.161216609892150207680474665447, 10.09972061078248011750852598771

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