Properties

Label 2-1232-1.1-c3-0-19
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  − 8.14·3-s + 18.9·5-s − 7·7-s + 39.3·9-s − 11·11-s − 53.8·13-s − 154.·15-s + 36.8·17-s − 1.35·19-s + 57.0·21-s − 26.7·23-s + 235.·25-s − 100.·27-s + 116.·29-s + 158.·31-s + 89.5·33-s − 132.·35-s + 162.·37-s + 438.·39-s − 454.·41-s − 384.·43-s + 745.·45-s + 264.·47-s + 49·49-s − 299.·51-s + 113.·53-s − 208.·55-s + ⋯
L(s)  = 1  − 1.56·3-s + 1.69·5-s − 0.377·7-s + 1.45·9-s − 0.301·11-s − 1.14·13-s − 2.65·15-s + 0.525·17-s − 0.0164·19-s + 0.592·21-s − 0.242·23-s + 1.88·25-s − 0.714·27-s + 0.743·29-s + 0.921·31-s + 0.472·33-s − 0.641·35-s + 0.722·37-s + 1.80·39-s − 1.72·41-s − 1.36·43-s + 2.47·45-s + 0.820·47-s + 0.142·49-s − 0.823·51-s + 0.295·53-s − 0.511·55-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

\(L(2)\) \(\approx\) \(1.340653939\)
\(L(\frac12)\) \(\approx\) \(1.340653939\)
\(L(\frac{5}{2})\) 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 \( 1 + 8.14T + 27T^{2} \)
5 \( 1 - 18.9T + 125T^{2} \)
13 \( 1 + 53.8T + 2.19e3T^{2} \)
17 \( 1 - 36.8T + 4.91e3T^{2} \)
19 \( 1 + 1.35T + 6.85e3T^{2} \)
23 \( 1 + 26.7T + 1.21e4T^{2} \)
29 \( 1 - 116.T + 2.43e4T^{2} \)
31 \( 1 - 158.T + 2.97e4T^{2} \)
37 \( 1 - 162.T + 5.06e4T^{2} \)
41 \( 1 + 454.T + 6.89e4T^{2} \)
43 \( 1 + 384.T + 7.95e4T^{2} \)
47 \( 1 - 264.T + 1.03e5T^{2} \)
53 \( 1 - 113.T + 1.48e5T^{2} \)
59 \( 1 - 602.T + 2.05e5T^{2} \)
61 \( 1 + 701.T + 2.26e5T^{2} \)
67 \( 1 + 422.T + 3.00e5T^{2} \)
71 \( 1 + 825.T + 3.57e5T^{2} \)
73 \( 1 + 121.T + 3.89e5T^{2} \)
79 \( 1 - 554.T + 4.93e5T^{2} \)
83 \( 1 - 880.T + 5.71e5T^{2} \)
89 \( 1 + 133.T + 7.04e5T^{2} \)
97 \( 1 - 1.37e3T + 9.12e5T^{2} \)
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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

−9.933703539020279624780236292525, −8.706463875755762274328695308426, −7.36702788319660462655250683876, −6.50295959067953189519516110858, −6.01547396505161128201761960896, −5.22035946008241386332621110684, −4.69715882602822512936731451130, −2.92609329028222240760142008694, −1.82050363114457992320327273307, −0.63973321353061769602489527740, 0.63973321353061769602489527740, 1.82050363114457992320327273307, 2.92609329028222240760142008694, 4.69715882602822512936731451130, 5.22035946008241386332621110684, 6.01547396505161128201761960896, 6.50295959067953189519516110858, 7.36702788319660462655250683876, 8.706463875755762274328695308426, 9.933703539020279624780236292525

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