Properties

Label 2-1216-1.1-c3-0-82
Degree $2$
Conductor $1216$
Sign $-1$
Analytic cond. $71.7463$
Root an. cond. $8.47032$
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  − 3-s + 21.2·5-s − 31.6·7-s − 26·9-s + 0.712·11-s + 38.9·13-s − 21.2·15-s − 54.1·17-s + 19·19-s + 31.6·21-s + 61.7·23-s + 328.·25-s + 53·27-s + 225.·29-s − 131.·31-s − 0.712·33-s − 673.·35-s − 94.1·37-s − 38.9·39-s − 108.·41-s − 205.·43-s − 553.·45-s − 523.·47-s + 658.·49-s + 54.1·51-s − 560.·53-s + 15.1·55-s + ⋯
L(s)  = 1  − 0.192·3-s + 1.90·5-s − 1.70·7-s − 0.962·9-s + 0.0195·11-s + 0.830·13-s − 0.366·15-s − 0.772·17-s + 0.229·19-s + 0.328·21-s + 0.560·23-s + 2.62·25-s + 0.377·27-s + 1.44·29-s − 0.762·31-s − 0.00375·33-s − 3.25·35-s − 0.418·37-s − 0.159·39-s − 0.414·41-s − 0.728·43-s − 1.83·45-s − 1.62·47-s + 1.91·49-s + 0.148·51-s − 1.45·53-s + 0.0371·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $-1$
Analytic conductor: \(71.7463\)
Root analytic conductor: \(8.47032\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1216} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1216,\ (\ :3/2),\ -1)\)

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 \( 1 \)
19 \( 1 - 19T \)
good3 \( 1 + T + 27T^{2} \)
5 \( 1 - 21.2T + 125T^{2} \)
7 \( 1 + 31.6T + 343T^{2} \)
11 \( 1 - 0.712T + 1.33e3T^{2} \)
13 \( 1 - 38.9T + 2.19e3T^{2} \)
17 \( 1 + 54.1T + 4.91e3T^{2} \)
23 \( 1 - 61.7T + 1.21e4T^{2} \)
29 \( 1 - 225.T + 2.43e4T^{2} \)
31 \( 1 + 131.T + 2.97e4T^{2} \)
37 \( 1 + 94.1T + 5.06e4T^{2} \)
41 \( 1 + 108.T + 6.89e4T^{2} \)
43 \( 1 + 205.T + 7.95e4T^{2} \)
47 \( 1 + 523.T + 1.03e5T^{2} \)
53 \( 1 + 560.T + 1.48e5T^{2} \)
59 \( 1 + 498.T + 2.05e5T^{2} \)
61 \( 1 + 179.T + 2.26e5T^{2} \)
67 \( 1 - 985.T + 3.00e5T^{2} \)
71 \( 1 + 904.T + 3.57e5T^{2} \)
73 \( 1 + 724.T + 3.89e5T^{2} \)
79 \( 1 + 1.16e3T + 4.93e5T^{2} \)
83 \( 1 - 1.13e3T + 5.71e5T^{2} \)
89 \( 1 - 208.T + 7.04e5T^{2} \)
97 \( 1 + 683.T + 9.12e5T^{2} \)
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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

−9.128769234310573620005228056931, −8.485242741495349942554915888680, −6.75128826479355224509431381338, −6.41848853858587837272007550388, −5.79728242906429228223159347915, −4.93072113533786590667549632860, −3.28167449050424977321876635221, −2.71232307156100525535306455302, −1.45031275962286229185646455701, 0, 1.45031275962286229185646455701, 2.71232307156100525535306455302, 3.28167449050424977321876635221, 4.93072113533786590667549632860, 5.79728242906429228223159347915, 6.41848853858587837272007550388, 6.75128826479355224509431381338, 8.485242741495349942554915888680, 9.128769234310573620005228056931

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