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Properties

Label 2-3072-8.5-c1-0-11
Degree $2$
Conductor $3072$
Sign $-i$
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 − 2.49i·5-s − 0.917·7-s − 9-s − 3.69i·11-s + 5.81i·13-s + 2.49·15-s − 0.867·17-s + 6.52i·19-s − 0.917i·21-s − 4·23-s − 1.23·25-s − i·27-s − 7.72i·29-s − 2.14·31-s + ⋯
L(s)  = 1  + 0.577i·3-s − 1.11i·5-s − 0.346·7-s − 0.333·9-s − 1.11i·11-s + 1.61i·13-s + 0.644·15-s − 0.210·17-s + 1.49i·19-s − 0.200i·21-s − 0.834·23-s − 0.246·25-s − 0.192i·27-s − 1.43i·29-s − 0.385·31-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3072\)    =    \(2^{10} \cdot 3\)
Sign: $-i$
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: \((2,\ 3072,\ (\ :1/2),\ -i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.115969675\)
\(L(\frac12)\) \(\approx\) \(1.115969675\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.49iT - 5T^{2} \)
7 \( 1 + 0.917T + 7T^{2} \)
11 \( 1 + 3.69iT - 11T^{2} \)
13 \( 1 - 5.81iT - 13T^{2} \)
17 \( 1 + 0.867T + 17T^{2} \)
19 \( 1 - 6.52iT - 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 + 7.72iT - 29T^{2} \)
31 \( 1 + 2.14T + 31T^{2} \)
37 \( 1 - 2.47iT - 37T^{2} \)
41 \( 1 - 9.58T + 41T^{2} \)
43 \( 1 - 9.58iT - 43T^{2} \)
47 \( 1 + 1.65T + 47T^{2} \)
53 \( 1 + 3.39iT - 53T^{2} \)
59 \( 1 - 12.7iT - 59T^{2} \)
61 \( 1 - 0.0231iT - 61T^{2} \)
67 \( 1 - 5.32iT - 67T^{2} \)
71 \( 1 + 11.8T + 71T^{2} \)
73 \( 1 - 15.2T + 73T^{2} \)
79 \( 1 - 8.40T + 79T^{2} \)
83 \( 1 - 1.96iT - 83T^{2} \)
89 \( 1 + 2.79T + 89T^{2} \)
97 \( 1 + 2.26T + 97T^{2} \)
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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

−8.927476604806854669852186350672, −8.319908745013508623013291378567, −7.62672083249077752322413571173, −6.20355103206431788945379398916, −6.04741367029341558667347494244, −4.90941705973011809561681955674, −4.19388089213539902429269071281, −3.60650405030778899220497870729, −2.28545103328228201773939431511, −1.11005660051757577003547248165, 0.37688068018701853824675008288, 1.97843630249707181363318995380, 2.81155323859836854211194757412, 3.47058496808738343570158224918, 4.72510686896404610636901826420, 5.57511927966567706076001952323, 6.44785298636714726320321091353, 7.10971456635908658194886986028, 7.49529333129988729351111758377, 8.395252693625359735862848204367

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