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Properties

Label 2-3072-48.29-c0-0-8
Degree $2$
Conductor $3072$
Sign $-0.382 + 0.923i$
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  + (−0.707 − 0.707i)3-s + 1.00i·9-s + (−1.41 − 1.41i)19-s − i·25-s + (0.707 − 0.707i)27-s + (1.41 − 1.41i)43-s + 49-s + 2.00i·57-s + (−1.41 − 1.41i)67-s − 2i·73-s + (−0.707 + 0.707i)75-s − 1.00·81-s − 2·97-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)3-s + 1.00i·9-s + (−1.41 − 1.41i)19-s − i·25-s + (0.707 − 0.707i)27-s + (1.41 − 1.41i)43-s + 49-s + 2.00i·57-s + (−1.41 − 1.41i)67-s − 2i·73-s + (−0.707 + 0.707i)75-s − 1.00·81-s − 2·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3072\)    =    \(2^{10} \cdot 3\)
Sign: $-0.382 + 0.923i$
Analytic conductor: \(1.53312\)
Root analytic conductor: \(1.23819\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3072} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3072,\ (\ :0),\ -0.382 + 0.923i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7304416220\)
\(L(\frac12)\) \(\approx\) \(0.7304416220\)
\(L(1)\) 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 \( 1 + (0.707 + 0.707i)T \)
good5 \( 1 + iT^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + iT^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + (1.41 + 1.41i)T + iT^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - iT^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (-1.41 + 1.41i)T - iT^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 + iT^{2} \)
67 \( 1 + (1.41 + 1.41i)T + iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + 2iT - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + 2T + T^{2} \)
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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

−8.624354187827936717032718184389, −7.79040635287870126992935266952, −7.05304989496546062757050854351, −6.42785414255528850372055491136, −5.75452046609404865511276035549, −4.81478377608992146015288375312, −4.16230907033899008918432432400, −2.72425410088202730598982268643, −1.95712338329204378325968463816, −0.51063105119599922290331032342, 1.36725538656361626900124952109, 2.74198755636911149032372621669, 3.89702041163973258740634789180, 4.32620683929509160746485781991, 5.43063973870499996027286541794, 5.94094145167146399279470691750, 6.71287564274634971063025448154, 7.61252048033858873517136605199, 8.501288829706252621221386772923, 9.188777729223045129511174371523

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