Properties

Label 2-1216-1.1-c3-0-18
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.227·3-s − 8.31·5-s − 8.08·7-s − 26.9·9-s − 12.7·11-s + 47.0·13-s − 1.89·15-s − 31.4·17-s + 19·19-s − 1.84·21-s + 19.0·23-s − 55.8·25-s − 12.3·27-s − 91.2·29-s − 293.·31-s − 2.91·33-s + 67.2·35-s − 215.·37-s + 10.7·39-s − 67.7·41-s + 308.·43-s + 224.·45-s − 108.·47-s − 277.·49-s − 7.17·51-s + 682.·53-s + 106.·55-s + ⋯
L(s)  = 1  + 0.0438·3-s − 0.743·5-s − 0.436·7-s − 0.998·9-s − 0.350·11-s + 1.00·13-s − 0.0326·15-s − 0.448·17-s + 0.229·19-s − 0.0191·21-s + 0.172·23-s − 0.446·25-s − 0.0876·27-s − 0.584·29-s − 1.70·31-s − 0.0153·33-s + 0.324·35-s − 0.958·37-s + 0.0440·39-s − 0.257·41-s + 1.09·43-s + 0.742·45-s − 0.337·47-s − 0.809·49-s − 0.0196·51-s + 1.76·53-s + 0.260·55-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9784548993\)
\(L(\frac12)\) \(\approx\) \(0.9784548993\)
\(L(\frac{5}{2})\) 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 \( 1 - 0.227T + 27T^{2} \)
5 \( 1 + 8.31T + 125T^{2} \)
7 \( 1 + 8.08T + 343T^{2} \)
11 \( 1 + 12.7T + 1.33e3T^{2} \)
13 \( 1 - 47.0T + 2.19e3T^{2} \)
17 \( 1 + 31.4T + 4.91e3T^{2} \)
23 \( 1 - 19.0T + 1.21e4T^{2} \)
29 \( 1 + 91.2T + 2.43e4T^{2} \)
31 \( 1 + 293.T + 2.97e4T^{2} \)
37 \( 1 + 215.T + 5.06e4T^{2} \)
41 \( 1 + 67.7T + 6.89e4T^{2} \)
43 \( 1 - 308.T + 7.95e4T^{2} \)
47 \( 1 + 108.T + 1.03e5T^{2} \)
53 \( 1 - 682.T + 1.48e5T^{2} \)
59 \( 1 + 250.T + 2.05e5T^{2} \)
61 \( 1 - 317.T + 2.26e5T^{2} \)
67 \( 1 - 940.T + 3.00e5T^{2} \)
71 \( 1 - 395.T + 3.57e5T^{2} \)
73 \( 1 - 975.T + 3.89e5T^{2} \)
79 \( 1 + 922.T + 4.93e5T^{2} \)
83 \( 1 + 1.16e3T + 5.71e5T^{2} \)
89 \( 1 - 685.T + 7.04e5T^{2} \)
97 \( 1 - 211.T + 9.12e5T^{2} \)
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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

−9.150603511959296760322710962791, −8.593488406694392387279716918853, −7.78038778777941103874677804663, −6.94130391336767487065544854205, −5.94547890690113992705228182606, −5.23147452779487324810306963110, −3.88634965986273758773453391693, −3.33954277348374511772005961027, −2.09126789578427975043930095052, −0.48303801987815796509169113044, 0.48303801987815796509169113044, 2.09126789578427975043930095052, 3.33954277348374511772005961027, 3.88634965986273758773453391693, 5.23147452779487324810306963110, 5.94547890690113992705228182606, 6.94130391336767487065544854205, 7.78038778777941103874677804663, 8.593488406694392387279716918853, 9.150603511959296760322710962791

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