# Properties

 Label 2-1216-1.1-c3-0-52 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 $1$

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## Dirichlet series

 L(s)  = 1 − 8.28·3-s − 2.38·5-s + 5.83·7-s + 41.6·9-s + 7.33·11-s − 55.6·13-s + 19.7·15-s − 10.0·17-s + 19·19-s − 48.3·21-s − 9.26·23-s − 119.·25-s − 121.·27-s + 83.9·29-s + 202.·31-s − 60.7·33-s − 13.9·35-s − 95.2·37-s + 460.·39-s − 25.9·41-s + 119.·43-s − 99.3·45-s + 467.·47-s − 308.·49-s + 83.1·51-s + 764.·53-s − 17.4·55-s + ⋯
 L(s)  = 1 − 1.59·3-s − 0.213·5-s + 0.315·7-s + 1.54·9-s + 0.201·11-s − 1.18·13-s + 0.340·15-s − 0.143·17-s + 0.229·19-s − 0.502·21-s − 0.0839·23-s − 0.954·25-s − 0.866·27-s + 0.537·29-s + 1.17·31-s − 0.320·33-s − 0.0671·35-s − 0.423·37-s + 1.89·39-s − 0.0990·41-s + 0.424·43-s − 0.329·45-s + 1.45·47-s − 0.900·49-s + 0.228·51-s + 1.98·53-s − 0.0428·55-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: $$1$$ 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 + 8.28T + 27T^{2}$$
5 $$1 + 2.38T + 125T^{2}$$
7 $$1 - 5.83T + 343T^{2}$$
11 $$1 - 7.33T + 1.33e3T^{2}$$
13 $$1 + 55.6T + 2.19e3T^{2}$$
17 $$1 + 10.0T + 4.91e3T^{2}$$
23 $$1 + 9.26T + 1.21e4T^{2}$$
29 $$1 - 83.9T + 2.43e4T^{2}$$
31 $$1 - 202.T + 2.97e4T^{2}$$
37 $$1 + 95.2T + 5.06e4T^{2}$$
41 $$1 + 25.9T + 6.89e4T^{2}$$
43 $$1 - 119.T + 7.95e4T^{2}$$
47 $$1 - 467.T + 1.03e5T^{2}$$
53 $$1 - 764.T + 1.48e5T^{2}$$
59 $$1 + 69.1T + 2.05e5T^{2}$$
61 $$1 - 398.T + 2.26e5T^{2}$$
67 $$1 - 243.T + 3.00e5T^{2}$$
71 $$1 + 781.T + 3.57e5T^{2}$$
73 $$1 + 711.T + 3.89e5T^{2}$$
79 $$1 - 723.T + 4.93e5T^{2}$$
83 $$1 - 1.22e3T + 5.71e5T^{2}$$
89 $$1 + 653.T + 7.04e5T^{2}$$
97 $$1 - 1.69e3T + 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

−9.072784639874110037254458708971, −7.920028212792913808888699565297, −7.14555099035619371390201214556, −6.37337248580923672697341728267, −5.50823922266162208765880663122, −4.82754795419624018468690951103, −4.01436924759045570102158335123, −2.43292877360622137588875974118, −1.03883609032757913509121252243, 0, 1.03883609032757913509121252243, 2.43292877360622137588875974118, 4.01436924759045570102158335123, 4.82754795419624018468690951103, 5.50823922266162208765880663122, 6.37337248580923672697341728267, 7.14555099035619371390201214556, 7.920028212792913808888699565297, 9.072784639874110037254458708971