↑

Properties

Label 2-3072-96.5-c0-0-4
Degree $2$
Conductor $3072$
Sign $-0.195 - 0.980i$
Analytic cond. $1.53312$
Root an. cond. $1.23819$
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  + (0.382 + 0.923i)3-s + (1.30 + 1.30i)7-s + (−0.707 + 0.707i)9-s + (−0.707 + 0.292i)13-s + (1.30 − 0.541i)19-s + (−0.707 + 1.70i)21-s + (0.707 + 0.707i)25-s + (−0.923 − 0.382i)27-s − 0.765·31-s + (−1.70 − 0.707i)37-s + (−0.541 − 0.541i)39-s + (0.541 − 1.30i)43-s + 2.41i·49-s + (1 + 0.999i)57-s + (0.292 + 0.707i)61-s + ⋯
L(s)  = 1  + (0.382 + 0.923i)3-s + (1.30 + 1.30i)7-s + (−0.707 + 0.707i)9-s + (−0.707 + 0.292i)13-s + (1.30 − 0.541i)19-s + (−0.707 + 1.70i)21-s + (0.707 + 0.707i)25-s + (−0.923 − 0.382i)27-s − 0.765·31-s + (−1.70 − 0.707i)37-s + (−0.541 − 0.541i)39-s + (0.541 − 1.30i)43-s + 2.41i·49-s + (1 + 0.999i)57-s + (0.292 + 0.707i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.195 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.195 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3072\)    =    \(2^{10} \cdot 3\)
Sign: $-0.195 - 0.980i$
Analytic conductor: \(1.53312\)
Root analytic conductor: \(1.23819\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3072} (2945, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3072,\ (\ :0),\ -0.195 - 0.980i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.561679168\)
\(L(\frac12)\) \(\approx\) \(1.561679168\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.382 - 0.923i)T \)
good5 \( 1 + (-0.707 - 0.707i)T^{2} \)
7 \( 1 + (-1.30 - 1.30i)T + iT^{2} \)
11 \( 1 + (0.707 + 0.707i)T^{2} \)
13 \( 1 + (0.707 - 0.292i)T + (0.707 - 0.707i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + (-1.30 + 0.541i)T + (0.707 - 0.707i)T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 + (0.707 - 0.707i)T^{2} \)
31 \( 1 + 0.765T + T^{2} \)
37 \( 1 + (1.70 + 0.707i)T + (0.707 + 0.707i)T^{2} \)
41 \( 1 + iT^{2} \)
43 \( 1 + (-0.541 + 1.30i)T + (-0.707 - 0.707i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (0.707 + 0.707i)T^{2} \)
59 \( 1 + (-0.707 - 0.707i)T^{2} \)
61 \( 1 + (-0.292 - 0.707i)T + (-0.707 + 0.707i)T^{2} \)
67 \( 1 + (-0.707 + 0.707i)T^{2} \)
71 \( 1 - iT^{2} \)
73 \( 1 + (-1 + i)T - iT^{2} \)
79 \( 1 + 0.765iT - T^{2} \)
83 \( 1 + (-0.707 + 0.707i)T^{2} \)
89 \( 1 - iT^{2} \)
97 \( 1 + 1.41T + T^{2} \)
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

−9.080886902501002199301950216951, −8.587436446293264854146410523025, −7.72533569856990462283669552770, −7.05487512839803985361418305699, −5.54340187220514545178722842960, −5.30890861605611675524864311932, −4.62180652329803961930987816429, −3.52941823085183219732624520022, −2.60193172671760953410885215598, −1.80085573098790808429731454552, 0.980619162143275163864283279250, 1.79215098797286723014965572334, 2.96690950421897460765673965633, 3.88312981101868358593417281620, 4.87692883410226176888206805073, 5.54853411043901451879377084549, 6.80261649397267575533617827695, 7.20353831260236142302726266911, 7.986942163888712893927802457173, 8.269406781265971448192523822986

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