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Properties

Label 2-3072-96.29-c0-0-6
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 + (1.30 − 1.30i)7-s + (0.707 + 0.707i)9-s + (0.292 − 0.707i)13-s + (0.541 − 1.30i)19-s + (−1.70 + 0.707i)21-s + (−0.707 + 0.707i)25-s + (−0.382 − 0.923i)27-s + 0.765·31-s + (0.707 + 1.70i)37-s + (−0.541 + 0.541i)39-s + (−1.30 + 0.541i)43-s − 2.41i·49-s + (−1 + 0.999i)57-s + (0.707 + 0.292i)61-s + ⋯
L(s)  = 1  + (−0.923 − 0.382i)3-s + (1.30 − 1.30i)7-s + (0.707 + 0.707i)9-s + (0.292 − 0.707i)13-s + (0.541 − 1.30i)19-s + (−1.70 + 0.707i)21-s + (−0.707 + 0.707i)25-s + (−0.382 − 0.923i)27-s + 0.765·31-s + (0.707 + 1.70i)37-s + (−0.541 + 0.541i)39-s + (−1.30 + 0.541i)43-s − 2.41i·49-s + (−1 + 0.999i)57-s + (0.707 + 0.292i)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: $\chi_{3072} (641, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3072,\ (\ :0),\ 0.195 + 0.980i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.081974895\)
\(L(\frac12)\) \(\approx\) \(1.081974895\)
\(L(1)\) 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 \( 1 + (-1.30 + 1.30i)T - iT^{2} \)
11 \( 1 + (-0.707 + 0.707i)T^{2} \)
13 \( 1 + (-0.292 + 0.707i)T + (-0.707 - 0.707i)T^{2} \)
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 \( 1 - 0.765T + T^{2} \)
37 \( 1 + (-0.707 - 1.70i)T + (-0.707 + 0.707i)T^{2} \)
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 \( 1 + (-0.707 - 0.292i)T + (0.707 + 0.707i)T^{2} \)
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 \( 1 + 1.41T + T^{2} \)
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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

−8.409960443717712875088211715039, −7.86593124119516729669967765815, −7.22510282726203949048534033816, −6.58651204656081720602391047158, −5.58387979466131251070812516145, −4.82849559737341741729958851297, −4.36283664209307485351491120774, −3.12299953769093333479382334896, −1.66989907582111133712343168679, −0.860012202235964090556655836363, 1.43064534959063924852575815594, 2.31842780243654169438872265015, 3.77517641070543661383595041347, 4.50829667785073276366896058865, 5.37515121913587532109360754552, 5.79835074883631151497514687103, 6.58309021507933079807992295538, 7.65787612362301787721560841354, 8.329068215652048172350978290375, 9.060051334885394777286508586732

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