↑

Properties

Label 2-960-960.749-c0-0-0
Degree $2$
Conductor $960$
Sign $0.956 - 0.290i$
Analytic cond. $0.479102$
Root an. cond. $0.692172$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.980 − 0.195i)2-s + (0.555 + 0.831i)3-s + (0.923 + 0.382i)4-s + (−0.195 − 0.980i)5-s + (−0.382 − 0.923i)6-s + (−0.831 − 0.555i)8-s + (−0.382 + 0.923i)9-s + i·10-s + (0.195 + 0.980i)12-s + (0.707 − 0.707i)15-s + (0.707 + 0.707i)16-s + (0.275 + 0.275i)17-s + (0.555 − 0.831i)18-s + (1.63 + 0.324i)19-s + (0.195 − 0.980i)20-s + ⋯
L(s)  = 1  + (−0.980 − 0.195i)2-s + (0.555 + 0.831i)3-s + (0.923 + 0.382i)4-s + (−0.195 − 0.980i)5-s + (−0.382 − 0.923i)6-s + (−0.831 − 0.555i)8-s + (−0.382 + 0.923i)9-s + i·10-s + (0.195 + 0.980i)12-s + (0.707 − 0.707i)15-s + (0.707 + 0.707i)16-s + (0.275 + 0.275i)17-s + (0.555 − 0.831i)18-s + (1.63 + 0.324i)19-s + (0.195 − 0.980i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 - 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 - 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $0.956 - 0.290i$
Analytic conductor: \(0.479102\)
Root analytic conductor: \(0.692172\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (749, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :0),\ 0.956 - 0.290i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7978637997\)
\(L(\frac12)\) \(\approx\) \(0.7978637997\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.980 + 0.195i)T \)
3 \( 1 + (-0.555 - 0.831i)T \)
5 \( 1 + (0.195 + 0.980i)T \)
good7 \( 1 + (0.707 - 0.707i)T^{2} \)
11 \( 1 + (-0.382 - 0.923i)T^{2} \)
13 \( 1 + (0.923 + 0.382i)T^{2} \)
17 \( 1 + (-0.275 - 0.275i)T + iT^{2} \)
19 \( 1 + (-1.63 - 0.324i)T + (0.923 + 0.382i)T^{2} \)
23 \( 1 + (-1.81 - 0.750i)T + (0.707 + 0.707i)T^{2} \)
29 \( 1 + (-0.382 + 0.923i)T^{2} \)
31 \( 1 + 1.84iT - T^{2} \)
37 \( 1 + (-0.923 + 0.382i)T^{2} \)
41 \( 1 + (-0.707 - 0.707i)T^{2} \)
43 \( 1 + (0.382 + 0.923i)T^{2} \)
47 \( 1 + (-0.785 - 0.785i)T + iT^{2} \)
53 \( 1 + (0.636 + 0.425i)T + (0.382 + 0.923i)T^{2} \)
59 \( 1 + (0.923 - 0.382i)T^{2} \)
61 \( 1 + (0.923 + 1.38i)T + (-0.382 + 0.923i)T^{2} \)
67 \( 1 + (0.382 - 0.923i)T^{2} \)
71 \( 1 + (0.707 - 0.707i)T^{2} \)
73 \( 1 + (0.707 + 0.707i)T^{2} \)
79 \( 1 + (1 - i)T - iT^{2} \)
83 \( 1 + (0.750 + 0.149i)T + (0.923 + 0.382i)T^{2} \)
89 \( 1 + (-0.707 + 0.707i)T^{2} \)
97 \( 1 + T^{2} \)
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

−9.830355363171089486992677814638, −9.488446231581033010177214032296, −8.825289050556513985928721813709, −7.893642125024167756872176641485, −7.45453305709658081889849087604, −5.87660358359649099803992227486, −4.98344910340118423411089077203, −3.79379137017845518994948433391, −2.88382182842019981485235886192, −1.37550894749488483039028189351, 1.25831427561184842027981259474, 2.73920380934427429150708416734, 3.24423136135569436242754514441, 5.25744728013731641641656664934, 6.40000431649447318163284165443, 7.15298692765118034563167672362, 7.45508604780163789903591873857, 8.545308367296151959735496569566, 9.174662474852540780993993240237, 10.09463205852604637222051790477

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