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Properties

Label 2-3072-96.53-c0-0-4
Degree $2$
Conductor $3072$
Sign $0.980 + 0.195i$
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

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

Invariants

Degree: \(2\)
Conductor: \(3072\)    =    \(2^{10} \cdot 3\)
Sign: $0.980 + 0.195i$
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: \((2,\ 3072,\ (\ :0),\ 0.980 + 0.195i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.979564039\)
\(L(\frac12)\) \(\approx\) \(1.979564039\)
\(L(1)\) 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

−8.642327537316654596639118865370, −8.249879046359708050269653833264, −7.62897941545409615275644997882, −6.68606116018552504526243649711, −5.83646152121157871078088098784, −4.92452119695903952817154050584, −4.27216517298089162966567514250, −2.84434572954066616875351683196, −2.45211177330195888799014914799, −1.37775257131098648125502965754, 1.48195947469730893533277205280, 2.17641774065398132498169722608, 3.60686790398562215294554786950, 4.14004276249653383808873034755, 4.75544782360024383378348569072, 5.76680267081975741805729133293, 7.01356097793226159697830928709, 7.58611244517919865645404142313, 8.057976516467526443312650081598, 8.868898757170220995823078228465

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