Properties

Label 2-960-15.14-c2-0-75
Degree $2$
Conductor $960$
Sign $0.505 + 0.862i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.938 + 2.84i)3-s + (4.88 − 1.05i)5-s − 6.81i·7-s + (−7.23 + 5.34i)9-s − 7.52i·11-s − 16.2i·13-s + (7.58 + 12.9i)15-s − 4.11·17-s + 7.86·19-s + (19.4 − 6.39i)21-s − 19.5·23-s + (22.7 − 10.2i)25-s + (−22.0 − 15.6i)27-s − 55.8i·29-s − 43.4·31-s + ⋯
L(s)  = 1  + (0.312 + 0.949i)3-s + (0.977 − 0.210i)5-s − 0.973i·7-s + (−0.804 + 0.594i)9-s − 0.684i·11-s − 1.24i·13-s + (0.505 + 0.862i)15-s − 0.242·17-s + 0.413·19-s + (0.924 − 0.304i)21-s − 0.848·23-s + (0.911 − 0.411i)25-s + (−0.815 − 0.578i)27-s − 1.92i·29-s − 1.40·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.505 + 0.862i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.505 + 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $0.505 + 0.862i$
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: \((2,\ 960,\ (\ :1),\ 0.505 + 0.862i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.004256462\)
\(L(\frac12)\) \(\approx\) \(2.004256462\)
\(L(2)\) 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 \( 1 + (-0.938 - 2.84i)T \)
5 \( 1 + (-4.88 + 1.05i)T \)
good7 \( 1 + 6.81iT - 49T^{2} \)
11 \( 1 + 7.52iT - 121T^{2} \)
13 \( 1 + 16.2iT - 169T^{2} \)
17 \( 1 + 4.11T + 289T^{2} \)
19 \( 1 - 7.86T + 361T^{2} \)
23 \( 1 + 19.5T + 529T^{2} \)
29 \( 1 + 55.8iT - 841T^{2} \)
31 \( 1 + 43.4T + 961T^{2} \)
37 \( 1 - 31.5iT - 1.36e3T^{2} \)
41 \( 1 + 51.3iT - 1.68e3T^{2} \)
43 \( 1 + 51.2iT - 1.84e3T^{2} \)
47 \( 1 + 61.7T + 2.20e3T^{2} \)
53 \( 1 - 82.7T + 2.80e3T^{2} \)
59 \( 1 - 97.6iT - 3.48e3T^{2} \)
61 \( 1 + 4.13T + 3.72e3T^{2} \)
67 \( 1 - 63.1iT - 4.48e3T^{2} \)
71 \( 1 + 40.3iT - 5.04e3T^{2} \)
73 \( 1 - 78.5iT - 5.32e3T^{2} \)
79 \( 1 - 51.0T + 6.24e3T^{2} \)
83 \( 1 + 2.72T + 6.88e3T^{2} \)
89 \( 1 + 70.4iT - 7.92e3T^{2} \)
97 \( 1 - 3.44iT - 9.40e3T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.966366046474626518923088272147, −8.927590525042551886642755323011, −8.213988843958934332723451779841, −7.26566065530984028875225423146, −5.90053626706429219747179662983, −5.44066683156005633329913020731, −4.26287771586326726042318715184, −3.40804117809498443905402645534, −2.25887780508512173697035919211, −0.57699426966219909820980286273, 1.65577918243954360396588034180, 2.16463086078835321516458355140, 3.33360860668950045369001052821, 4.91897252583588334568590038238, 5.83817895983153236045465237784, 6.61464783759057584352017877974, 7.26125768891504669499892247748, 8.395533086170699002896380756181, 9.244731361872997631227475843572, 9.570418797787614515446734884018

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