Properties

Label 2-3072-8.5-c1-0-13
Degree $2$
Conductor $3072$
Sign $-1$
Analytic cond. $24.5300$
Root an. cond. $4.95278$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·3-s + 3.79i·5-s + 2.15·7-s − 9-s − 2.54i·11-s + 1.95i·13-s − 3.79·15-s + 0.224·17-s − 0.224i·19-s + 2.15i·21-s − 2.82·23-s − 9.42·25-s i·27-s + 2.62i·29-s − 1.84·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + 1.69i·5-s + 0.816·7-s − 0.333·9-s − 0.766i·11-s + 0.542i·13-s − 0.980·15-s + 0.0545·17-s − 0.0515i·19-s + 0.471i·21-s − 0.589·23-s − 1.88·25-s − 0.192i·27-s + 0.487i·29-s − 0.330·31-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3072\)    =    \(2^{10} \cdot 3\)
Sign: $-1$
Analytic conductor: \(24.5300\)
Root analytic conductor: \(4.95278\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3072} (1537, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3072,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.308667440\)
\(L(\frac12)\) \(\approx\) \(1.308667440\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
good5 \( 1 - 3.79iT - 5T^{2} \)
7 \( 1 - 2.15T + 7T^{2} \)
11 \( 1 + 2.54iT - 11T^{2} \)
13 \( 1 - 1.95iT - 13T^{2} \)
17 \( 1 - 0.224T + 17T^{2} \)
19 \( 1 + 0.224iT - 19T^{2} \)
23 \( 1 + 2.82T + 23T^{2} \)
29 \( 1 - 2.62iT - 29T^{2} \)
31 \( 1 + 1.84T + 31T^{2} \)
37 \( 1 - 5.18iT - 37T^{2} \)
41 \( 1 + 5.88T + 41T^{2} \)
43 \( 1 - 10.9iT - 43T^{2} \)
47 \( 1 + 2.82T + 47T^{2} \)
53 \( 1 - 10.6iT - 53T^{2} \)
59 \( 1 - 5.65iT - 59T^{2} \)
61 \( 1 + 8.46iT - 61T^{2} \)
67 \( 1 + 14.7iT - 67T^{2} \)
71 \( 1 + 4.31T + 71T^{2} \)
73 \( 1 + 5.97T + 73T^{2} \)
79 \( 1 - 15.0T + 79T^{2} \)
83 \( 1 - 14.3iT - 83T^{2} \)
89 \( 1 - 1.42T + 89T^{2} \)
97 \( 1 + 16.3T + 97T^{2} \)
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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

−9.177730645240211616917209939174, −8.185384706701491511704366747146, −7.69376601574364777251167385240, −6.67011802317632519784179953800, −6.22354838497288595376339731032, −5.24234261776260177001516539177, −4.33693633064703629021155670864, −3.42082400333398484311204239755, −2.80644089449937185565487791420, −1.67913195417614797231818087898, 0.39273111951828374290520963555, 1.51714152573697640324214817255, 2.16080802610381799170493785515, 3.73233079738001389071818185934, 4.59599161843303560748582141099, 5.24348176327349365345844026396, 5.81906555822177007500412359363, 7.00270339143941557632655921950, 7.75229006938247099425341451592, 8.342765436381654551342806235905

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