Properties

Label 2-1232-1.1-c3-0-16
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  − 1.12·3-s + 10.7·5-s − 7·7-s − 25.7·9-s − 11·11-s − 81.4·13-s − 12.0·15-s − 32.4·17-s − 22.1·19-s + 7.89·21-s + 105.·23-s − 9.88·25-s + 59.4·27-s − 208.·29-s + 229.·31-s + 12.4·33-s − 75.1·35-s + 195.·37-s + 91.7·39-s + 482.·41-s + 255.·43-s − 276.·45-s − 155.·47-s + 49·49-s + 36.5·51-s + 482.·53-s − 118.·55-s + ⋯
L(s)  = 1  − 0.216·3-s + 0.959·5-s − 0.377·7-s − 0.952·9-s − 0.301·11-s − 1.73·13-s − 0.208·15-s − 0.462·17-s − 0.267·19-s + 0.0819·21-s + 0.958·23-s − 0.0790·25-s + 0.423·27-s − 1.33·29-s + 1.33·31-s + 0.0654·33-s − 0.362·35-s + 0.869·37-s + 0.376·39-s + 1.83·41-s + 0.907·43-s − 0.914·45-s − 0.481·47-s + 0.142·49-s + 0.100·51-s + 1.25·53-s − 0.289·55-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

\(L(2)\) \(\approx\) \(1.463607993\)
\(L(\frac12)\) \(\approx\) \(1.463607993\)
\(L(\frac{5}{2})\) 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 \( 1 + 1.12T + 27T^{2} \)
5 \( 1 - 10.7T + 125T^{2} \)
13 \( 1 + 81.4T + 2.19e3T^{2} \)
17 \( 1 + 32.4T + 4.91e3T^{2} \)
19 \( 1 + 22.1T + 6.85e3T^{2} \)
23 \( 1 - 105.T + 1.21e4T^{2} \)
29 \( 1 + 208.T + 2.43e4T^{2} \)
31 \( 1 - 229.T + 2.97e4T^{2} \)
37 \( 1 - 195.T + 5.06e4T^{2} \)
41 \( 1 - 482.T + 6.89e4T^{2} \)
43 \( 1 - 255.T + 7.95e4T^{2} \)
47 \( 1 + 155.T + 1.03e5T^{2} \)
53 \( 1 - 482.T + 1.48e5T^{2} \)
59 \( 1 - 656.T + 2.05e5T^{2} \)
61 \( 1 - 162.T + 2.26e5T^{2} \)
67 \( 1 - 359.T + 3.00e5T^{2} \)
71 \( 1 - 318.T + 3.57e5T^{2} \)
73 \( 1 + 294.T + 3.89e5T^{2} \)
79 \( 1 + 590.T + 4.93e5T^{2} \)
83 \( 1 + 153.T + 5.71e5T^{2} \)
89 \( 1 + 696.T + 7.04e5T^{2} \)
97 \( 1 + 1.86e3T + 9.12e5T^{2} \)
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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

−9.516951653478252509604783147313, −8.674006006116080622105029332173, −7.62193412907415170995712952945, −6.80127420900655699430019449388, −5.85344096068421162179684229865, −5.31875903930345737465016923913, −4.29071108640690567179418710344, −2.74813682840099149412317275962, −2.30207858446712056223111415138, −0.59093956876685816823265618532, 0.59093956876685816823265618532, 2.30207858446712056223111415138, 2.74813682840099149412317275962, 4.29071108640690567179418710344, 5.31875903930345737465016923913, 5.85344096068421162179684229865, 6.80127420900655699430019449388, 7.62193412907415170995712952945, 8.674006006116080622105029332173, 9.516951653478252509604783147313

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