Properties

Label 2-1232-1.1-c3-0-47
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  − 5.70·3-s − 8.47·5-s + 7·7-s + 5.58·9-s − 11·11-s + 29.0·13-s + 48.3·15-s + 18.9·17-s − 71.0·19-s − 39.9·21-s − 57.0·23-s − 53.2·25-s + 122.·27-s + 174.·29-s − 98.8·31-s + 62.7·33-s − 59.3·35-s + 352.·37-s − 165.·39-s + 294.·41-s − 17.7·43-s − 47.3·45-s + 233.·47-s + 49·49-s − 107.·51-s + 225.·53-s + 93.1·55-s + ⋯
L(s)  = 1  − 1.09·3-s − 0.757·5-s + 0.377·7-s + 0.206·9-s − 0.301·11-s + 0.618·13-s + 0.832·15-s + 0.269·17-s − 0.857·19-s − 0.415·21-s − 0.517·23-s − 0.425·25-s + 0.871·27-s + 1.11·29-s − 0.572·31-s + 0.331·33-s − 0.286·35-s + 1.56·37-s − 0.679·39-s + 1.12·41-s − 0.0628·43-s − 0.156·45-s + 0.724·47-s + 0.142·49-s − 0.296·51-s + 0.584·53-s + 0.228·55-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: \(1\)
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 \( 1 + 5.70T + 27T^{2} \)
5 \( 1 + 8.47T + 125T^{2} \)
13 \( 1 - 29.0T + 2.19e3T^{2} \)
17 \( 1 - 18.9T + 4.91e3T^{2} \)
19 \( 1 + 71.0T + 6.85e3T^{2} \)
23 \( 1 + 57.0T + 1.21e4T^{2} \)
29 \( 1 - 174.T + 2.43e4T^{2} \)
31 \( 1 + 98.8T + 2.97e4T^{2} \)
37 \( 1 - 352.T + 5.06e4T^{2} \)
41 \( 1 - 294.T + 6.89e4T^{2} \)
43 \( 1 + 17.7T + 7.95e4T^{2} \)
47 \( 1 - 233.T + 1.03e5T^{2} \)
53 \( 1 - 225.T + 1.48e5T^{2} \)
59 \( 1 + 299.T + 2.05e5T^{2} \)
61 \( 1 - 760.T + 2.26e5T^{2} \)
67 \( 1 - 173.T + 3.00e5T^{2} \)
71 \( 1 + 1.01e3T + 3.57e5T^{2} \)
73 \( 1 + 612.T + 3.89e5T^{2} \)
79 \( 1 - 34.8T + 4.93e5T^{2} \)
83 \( 1 - 188.T + 5.71e5T^{2} \)
89 \( 1 + 1.09e3T + 7.04e5T^{2} \)
97 \( 1 - 285.T + 9.12e5T^{2} \)
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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

−8.766682748563349426969670543923, −8.097049928413454484847070384127, −7.28692272837988702616436077594, −6.22311741924035161602309228244, −5.68962961358465638430633337136, −4.61738634343120990046799781716, −3.92947039733666461026610878244, −2.56342755547118177495019595074, −1.04459560422884113672584049805, 0, 1.04459560422884113672584049805, 2.56342755547118177495019595074, 3.92947039733666461026610878244, 4.61738634343120990046799781716, 5.68962961358465638430633337136, 6.22311741924035161602309228244, 7.28692272837988702616436077594, 8.097049928413454484847070384127, 8.766682748563349426969670543923

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