Properties

Label 2-1232-1.1-c3-0-32
Degree $2$
Conductor $1232$
Sign $1$
Analytic cond. $72.6903$
Root an. cond. $8.52586$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8.08·3-s + 21.9·5-s − 7·7-s + 38.2·9-s + 11·11-s + 16.3·13-s − 177.·15-s − 35.5·17-s + 64.1·19-s + 56.5·21-s + 141.·23-s + 358.·25-s − 91.2·27-s − 268.·29-s + 191.·31-s − 88.8·33-s − 153.·35-s − 101.·37-s − 132.·39-s + 250.·41-s + 523.·43-s + 841.·45-s − 609.·47-s + 49·49-s + 287.·51-s + 0.896·53-s + 241.·55-s + ⋯
L(s)  = 1  − 1.55·3-s + 1.96·5-s − 0.377·7-s + 1.41·9-s + 0.301·11-s + 0.349·13-s − 3.05·15-s − 0.506·17-s + 0.774·19-s + 0.587·21-s + 1.28·23-s + 2.86·25-s − 0.650·27-s − 1.72·29-s + 1.10·31-s − 0.468·33-s − 0.743·35-s − 0.452·37-s − 0.543·39-s + 0.954·41-s + 1.85·43-s + 2.78·45-s − 1.89·47-s + 0.142·49-s + 0.788·51-s + 0.00232·53-s + 0.592·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1232\)    =    \(2^{4} \cdot 7 \cdot 11\)
Sign: $1$
Analytic conductor: \(72.6903\)
Root analytic conductor: \(8.52586\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1232,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.892301929\)
\(L(\frac12)\) \(\approx\) \(1.892301929\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + 7T \)
11 \( 1 - 11T \)
good3 \( 1 + 8.08T + 27T^{2} \)
5 \( 1 - 21.9T + 125T^{2} \)
13 \( 1 - 16.3T + 2.19e3T^{2} \)
17 \( 1 + 35.5T + 4.91e3T^{2} \)
19 \( 1 - 64.1T + 6.85e3T^{2} \)
23 \( 1 - 141.T + 1.21e4T^{2} \)
29 \( 1 + 268.T + 2.43e4T^{2} \)
31 \( 1 - 191.T + 2.97e4T^{2} \)
37 \( 1 + 101.T + 5.06e4T^{2} \)
41 \( 1 - 250.T + 6.89e4T^{2} \)
43 \( 1 - 523.T + 7.95e4T^{2} \)
47 \( 1 + 609.T + 1.03e5T^{2} \)
53 \( 1 - 0.896T + 1.48e5T^{2} \)
59 \( 1 + 398.T + 2.05e5T^{2} \)
61 \( 1 + 327.T + 2.26e5T^{2} \)
67 \( 1 + 25.5T + 3.00e5T^{2} \)
71 \( 1 + 200.T + 3.57e5T^{2} \)
73 \( 1 - 745.T + 3.89e5T^{2} \)
79 \( 1 + 654.T + 4.93e5T^{2} \)
83 \( 1 + 1.02e3T + 5.71e5T^{2} \)
89 \( 1 + 13.7T + 7.04e5T^{2} \)
97 \( 1 - 868.T + 9.12e5T^{2} \)
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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

−9.485372536447798441847349476342, −8.945348448984859884557867371592, −7.28609075673118262469550627980, −6.49960320314823607965694670068, −5.97122083713372909214896409613, −5.38480106674274321644834777653, −4.58209417131968175394741521791, −2.97916955278849454282043775851, −1.70040444022063520283452949374, −0.803206283359199072399585794567, 0.803206283359199072399585794567, 1.70040444022063520283452949374, 2.97916955278849454282043775851, 4.58209417131968175394741521791, 5.38480106674274321644834777653, 5.97122083713372909214896409613, 6.49960320314823607965694670068, 7.28609075673118262469550627980, 8.945348448984859884557867371592, 9.485372536447798441847349476342

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