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Properties

Label 2-3072-48.29-c0-0-3
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

\(L(\frac{1}{2})\) \(\approx\) \(1.604183966\)
\(L(\frac12)\) \(\approx\) \(1.604183966\)
\(L(1)\) 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

−9.073221786252299923806390063414, −8.200111280201559902131618305349, −7.83501986510574242241607440647, −6.87203775467530392545073529481, −5.84708554138472942478538196571, −5.12709580101326776447169180230, −4.24377471230616338001811570280, −3.48088156479517302531132840367, −2.68939251459845200889645947595, −1.53350743937574894770540309698, 0.994218263864976680754827617339, 2.15055032143309626007799534134, 3.06496364151215791319726696955, 3.77662218771910359513202118686, 4.98295731119253467662722568788, 5.70008998277433213533701522031, 6.89726913440852268164912887252, 7.10191604597371451993263308553, 8.001015597769676503500658238449, 8.723918046626110272180542179297

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