Properties

Label 2-3072-8.5-c1-0-36
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 + 2.47i·5-s − 2.55·7-s − 9-s + 0.669i·11-s − 4.08i·13-s − 2.47·15-s + 6.44·17-s − 6.44i·19-s − 2.55i·21-s + 2.82·23-s − 1.11·25-s i·27-s − 4.35i·29-s − 6.55·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + 1.10i·5-s − 0.966·7-s − 0.333·9-s + 0.201i·11-s − 1.13i·13-s − 0.638·15-s + 1.56·17-s − 1.47i·19-s − 0.558i·21-s + 0.589·23-s − 0.223·25-s − 0.192i·27-s − 0.808i·29-s − 1.17·31-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

\(L(1)\) \(\approx\) \(1.427841727\)
\(L(\frac12)\) \(\approx\) \(1.427841727\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.47iT - 5T^{2} \)
7 \( 1 + 2.55T + 7T^{2} \)
11 \( 1 - 0.669iT - 11T^{2} \)
13 \( 1 + 4.08iT - 13T^{2} \)
17 \( 1 - 6.44T + 17T^{2} \)
19 \( 1 + 6.44iT - 19T^{2} \)
23 \( 1 - 2.82T + 23T^{2} \)
29 \( 1 + 4.35iT - 29T^{2} \)
31 \( 1 + 6.55T + 31T^{2} \)
37 \( 1 + 3.85iT - 37T^{2} \)
41 \( 1 + 0.788T + 41T^{2} \)
43 \( 1 + 0.550iT - 43T^{2} \)
47 \( 1 - 2.82T + 47T^{2} \)
53 \( 1 - 3.64iT - 53T^{2} \)
59 \( 1 + 5.65iT - 59T^{2} \)
61 \( 1 + 6.20iT - 61T^{2} \)
67 \( 1 - 2.99iT - 67T^{2} \)
71 \( 1 - 5.11T + 71T^{2} \)
73 \( 1 - 14.7T + 73T^{2} \)
79 \( 1 + 6.31T + 79T^{2} \)
83 \( 1 - 0.907iT - 83T^{2} \)
89 \( 1 - 6.31T + 89T^{2} \)
97 \( 1 - 12.6T + 97T^{2} \)
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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

−8.864044449426771805751739969976, −7.80143472639530778533690494164, −7.21104565495802094867417739106, −6.45344605186764234458029706025, −5.66690120220221941689534838957, −4.94839457235190040967003709317, −3.64390511414266871370337933563, −3.19439322986092907175846329698, −2.46416384824110385239545507976, −0.55100458888902263295292373052, 0.976980473448459421816177034803, 1.79164628328548991378238026534, 3.19171548592230284062624845366, 3.85470015373742224190889269354, 5.01300175062752681504219918948, 5.67879887776666989884640320044, 6.43112059276035337046748844864, 7.22731956880204589860313973094, 7.980493157110369153245819827599, 8.744884202914717085420417432723

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