Properties

Label 2-1232-1.1-c3-0-64
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.77·3-s + 1.84·5-s + 7·7-s − 19.3·9-s + 11·11-s + 24.6·13-s − 5.11·15-s + 17.8·17-s − 32.1·19-s − 19.4·21-s − 14.1·23-s − 121.·25-s + 128.·27-s − 41.5·29-s − 175.·31-s − 30.5·33-s + 12.9·35-s + 292.·37-s − 68.3·39-s + 154.·41-s + 277.·43-s − 35.6·45-s + 52.1·47-s + 49·49-s − 49.5·51-s + 82.3·53-s + 20.2·55-s + ⋯
L(s)  = 1  − 0.533·3-s + 0.164·5-s + 0.377·7-s − 0.714·9-s + 0.301·11-s + 0.525·13-s − 0.0880·15-s + 0.255·17-s − 0.388·19-s − 0.201·21-s − 0.128·23-s − 0.972·25-s + 0.915·27-s − 0.266·29-s − 1.01·31-s − 0.160·33-s + 0.0623·35-s + 1.30·37-s − 0.280·39-s + 0.587·41-s + 0.982·43-s − 0.117·45-s + 0.161·47-s + 0.142·49-s − 0.136·51-s + 0.213·53-s + 0.0497·55-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: $\chi_{1232} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1232,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 + 2.77T + 27T^{2} \)
5 \( 1 - 1.84T + 125T^{2} \)
13 \( 1 - 24.6T + 2.19e3T^{2} \)
17 \( 1 - 17.8T + 4.91e3T^{2} \)
19 \( 1 + 32.1T + 6.85e3T^{2} \)
23 \( 1 + 14.1T + 1.21e4T^{2} \)
29 \( 1 + 41.5T + 2.43e4T^{2} \)
31 \( 1 + 175.T + 2.97e4T^{2} \)
37 \( 1 - 292.T + 5.06e4T^{2} \)
41 \( 1 - 154.T + 6.89e4T^{2} \)
43 \( 1 - 277.T + 7.95e4T^{2} \)
47 \( 1 - 52.1T + 1.03e5T^{2} \)
53 \( 1 - 82.3T + 1.48e5T^{2} \)
59 \( 1 + 712.T + 2.05e5T^{2} \)
61 \( 1 + 647.T + 2.26e5T^{2} \)
67 \( 1 + 260.T + 3.00e5T^{2} \)
71 \( 1 + 369.T + 3.57e5T^{2} \)
73 \( 1 - 1.14e3T + 3.89e5T^{2} \)
79 \( 1 + 488.T + 4.93e5T^{2} \)
83 \( 1 + 548.T + 5.71e5T^{2} \)
89 \( 1 - 105.T + 7.04e5T^{2} \)
97 \( 1 + 1.36e3T + 9.12e5T^{2} \)
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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

−8.980602529817914538812331368092, −8.111062501730518771677325312234, −7.32914971355681737678481346131, −6.08191703891784493496339746517, −5.82315993035821251812876127539, −4.68583120707649640975149808911, −3.73679660251847105435503951111, −2.51842369260886060443881656025, −1.30457310507710787551504086594, 0, 1.30457310507710787551504086594, 2.51842369260886060443881656025, 3.73679660251847105435503951111, 4.68583120707649640975149808911, 5.82315993035821251812876127539, 6.08191703891784493496339746517, 7.32914971355681737678481346131, 8.111062501730518771677325312234, 8.980602529817914538812331368092

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