Properties

Label 2-3072-1.1-c1-0-32
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 yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 2.47·5-s + 2.55·7-s + 9-s + 0.669·11-s + 4.08·13-s − 2.47·15-s + 6.44·17-s + 6.44·19-s − 2.55·21-s − 2.82·23-s + 1.11·25-s − 27-s + 4.35·29-s − 6.55·31-s − 0.669·33-s + 6.32·35-s − 3.85·37-s − 4.08·39-s + 0.788·41-s − 0.550·43-s + 2.47·45-s + 2.82·47-s − 0.458·49-s − 6.44·51-s + 3.64·53-s + 1.65·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.10·5-s + 0.966·7-s + 0.333·9-s + 0.201·11-s + 1.13·13-s − 0.638·15-s + 1.56·17-s + 1.47·19-s − 0.558·21-s − 0.589·23-s + 0.223·25-s − 0.192·27-s + 0.808·29-s − 1.17·31-s − 0.116·33-s + 1.06·35-s − 0.634·37-s − 0.653·39-s + 0.123·41-s − 0.0840·43-s + 0.368·45-s + 0.412·47-s − 0.0654·49-s − 0.902·51-s + 0.500·53-s + 0.223·55-s + ⋯

Functional equation

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

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: $\chi_{3072} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3072,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.551636185\)
\(L(\frac12)\) \(\approx\) \(2.551636185\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
good5 \( 1 - 2.47T + 5T^{2} \)
7 \( 1 - 2.55T + 7T^{2} \)
11 \( 1 - 0.669T + 11T^{2} \)
13 \( 1 - 4.08T + 13T^{2} \)
17 \( 1 - 6.44T + 17T^{2} \)
19 \( 1 - 6.44T + 19T^{2} \)
23 \( 1 + 2.82T + 23T^{2} \)
29 \( 1 - 4.35T + 29T^{2} \)
31 \( 1 + 6.55T + 31T^{2} \)
37 \( 1 + 3.85T + 37T^{2} \)
41 \( 1 - 0.788T + 41T^{2} \)
43 \( 1 + 0.550T + 43T^{2} \)
47 \( 1 - 2.82T + 47T^{2} \)
53 \( 1 - 3.64T + 53T^{2} \)
59 \( 1 + 5.65T + 59T^{2} \)
61 \( 1 - 6.20T + 61T^{2} \)
67 \( 1 + 2.99T + 67T^{2} \)
71 \( 1 + 5.11T + 71T^{2} \)
73 \( 1 + 14.7T + 73T^{2} \)
79 \( 1 + 6.31T + 79T^{2} \)
83 \( 1 + 0.907T + 83T^{2} \)
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

−8.755764470722570785105404986937, −7.894709752441811289325244825152, −7.23613057087574792554541066923, −6.19148592311634122342441078321, −5.59779024063240824485900491089, −5.19114080822899924705615454903, −4.04897279529944352095370893819, −3.08613104818524905276246437440, −1.69524908028366097373143870760, −1.16694361824026132816022090914, 1.16694361824026132816022090914, 1.69524908028366097373143870760, 3.08613104818524905276246437440, 4.04897279529944352095370893819, 5.19114080822899924705615454903, 5.59779024063240824485900491089, 6.19148592311634122342441078321, 7.23613057087574792554541066923, 7.894709752441811289325244825152, 8.755764470722570785105404986937

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