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Properties

Label 2-3072-96.53-c0-0-3
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.923 + 0.382i)3-s + (0.541 + 0.541i)7-s + (0.707 − 0.707i)9-s + (−0.707 − 1.70i)13-s + (−0.541 − 1.30i)19-s + (−0.707 − 0.292i)21-s + (−0.707 − 0.707i)25-s + (−0.382 + 0.923i)27-s − 1.84·31-s + (0.292 − 0.707i)37-s + (1.30 + 1.30i)39-s + (1.30 + 0.541i)43-s − 0.414i·49-s + (1 + 0.999i)57-s + (−1.70 + 0.707i)61-s + ⋯
L(s)  = 1  + (−0.923 + 0.382i)3-s + (0.541 + 0.541i)7-s + (0.707 − 0.707i)9-s + (−0.707 − 1.70i)13-s + (−0.541 − 1.30i)19-s + (−0.707 − 0.292i)21-s + (−0.707 − 0.707i)25-s + (−0.382 + 0.923i)27-s − 1.84·31-s + (0.292 − 0.707i)37-s + (1.30 + 1.30i)39-s + (1.30 + 0.541i)43-s − 0.414i·49-s + (1 + 0.999i)57-s + (−1.70 + 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: $\chi_{3072} (1409, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3072,\ (\ :0),\ 0.195 + 0.980i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6612379144\)
\(L(\frac12)\) \(\approx\) \(0.6612379144\)
\(L(1)\) 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 \( 1 + (0.923 - 0.382i)T \)
good5 \( 1 + (0.707 + 0.707i)T^{2} \)
7 \( 1 + (-0.541 - 0.541i)T + iT^{2} \)
11 \( 1 + (-0.707 - 0.707i)T^{2} \)
13 \( 1 + (0.707 + 1.70i)T + (-0.707 + 0.707i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + (0.541 + 1.30i)T + (-0.707 + 0.707i)T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 + (-0.707 + 0.707i)T^{2} \)
31 \( 1 + 1.84T + T^{2} \)
37 \( 1 + (-0.292 + 0.707i)T + (-0.707 - 0.707i)T^{2} \)
41 \( 1 + iT^{2} \)
43 \( 1 + (-1.30 - 0.541i)T + (0.707 + 0.707i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (-0.707 - 0.707i)T^{2} \)
59 \( 1 + (0.707 + 0.707i)T^{2} \)
61 \( 1 + (1.70 - 0.707i)T + (0.707 - 0.707i)T^{2} \)
67 \( 1 + (0.707 - 0.707i)T^{2} \)
71 \( 1 - iT^{2} \)
73 \( 1 + (-1 + i)T - iT^{2} \)
79 \( 1 + 1.84iT - T^{2} \)
83 \( 1 + (0.707 - 0.707i)T^{2} \)
89 \( 1 - iT^{2} \)
97 \( 1 - 1.41T + T^{2} \)
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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

−8.895892851277243613738503742224, −7.81232223906855716815647088265, −7.32209781746216371024442291475, −6.22390029367628130793380713291, −5.60063527218153162192022841988, −4.99709570103875560136797441813, −4.24243479304154593103118792319, −3.10576577067648329037318044706, −2.06140395269720501797867106065, −0.45723127998124132989411076593, 1.47189524841912003188277277795, 2.11571277452877078173072909544, 3.84853687803841724122622008284, 4.39550369548282577741616021877, 5.28190116925457677210665824538, 6.03725789322019750956473829842, 6.85355828196574634868640017405, 7.44665924461593824074960436587, 8.048065270817418798924815667615, 9.197234810703116419437091083555

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