# Properties

 Label 2-1216-152.75-c1-0-33 Degree $2$ Conductor $1216$ Sign $-0.610 + 0.791i$ Analytic cond. $9.70980$ Root an. cond. $3.11605$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + 2.64i·3-s − 3.46i·5-s − i·7-s − 4.00·9-s − 2.64·13-s + 9.16·15-s − 3·17-s + (−3.46 + 2.64i)19-s + 2.64·21-s − 3i·23-s − 6.99·25-s − 2.64i·27-s − 7.93·29-s − 3.46·35-s − 10.5·37-s + ⋯
 L(s)  = 1 + 1.52i·3-s − 1.54i·5-s − 0.377i·7-s − 1.33·9-s − 0.733·13-s + 2.36·15-s − 0.727·17-s + (−0.794 + 0.606i)19-s + 0.577·21-s − 0.625i·23-s − 1.39·25-s − 0.509i·27-s − 1.47·29-s − 0.585·35-s − 1.73·37-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.610 + 0.791i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.610 + 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1216$$    =    $$2^{6} \cdot 19$$ Sign: $-0.610 + 0.791i$ Analytic conductor: $$9.70980$$ Root analytic conductor: $$3.11605$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{1216} (607, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1216,\ (\ :1/2),\ -0.610 + 0.791i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.3170006314$$ $$L(\frac12)$$ $$\approx$$ $$0.3170006314$$ $$L(\frac{3}{2})$$ 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 + (3.46 - 2.64i)T$$
good3 $$1 - 2.64iT - 3T^{2}$$
5 $$1 + 3.46iT - 5T^{2}$$
7 $$1 + iT - 7T^{2}$$
11 $$1 + 11T^{2}$$
13 $$1 + 2.64T + 13T^{2}$$
17 $$1 + 3T + 17T^{2}$$
23 $$1 + 3iT - 23T^{2}$$
29 $$1 + 7.93T + 29T^{2}$$
31 $$1 + 31T^{2}$$
37 $$1 + 10.5T + 37T^{2}$$
41 $$1 + 9.16iT - 41T^{2}$$
43 $$1 + 10.3T + 43T^{2}$$
47 $$1 - 47T^{2}$$
53 $$1 - 7.93T + 53T^{2}$$
59 $$1 - 7.93iT - 59T^{2}$$
61 $$1 - 6.92iT - 61T^{2}$$
67 $$1 + 13.2iT - 67T^{2}$$
71 $$1 + 9.16T + 71T^{2}$$
73 $$1 - 7T + 73T^{2}$$
79 $$1 - 9.16T + 79T^{2}$$
83 $$1 - 17.3T + 83T^{2}$$
89 $$1 - 9.16iT - 89T^{2}$$
97 $$1 + 9.16iT - 97T^{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

−9.264196976272009561057411200542, −8.942214318748438942151984420444, −8.165417637903947135280337490568, −6.98671653842948013868050678251, −5.61227935889091269545651207411, −5.02010597108753288798599096951, −4.28647876222509420141458533245, −3.67219907490891301998952603979, −1.98163288042046773655853772138, −0.12294204349789222863416519535, 1.91414723239338333977799859290, 2.51939043999563612524244591146, 3.58103567637207080362136103586, 5.17328779298672573431597533094, 6.26849212966148570758810072498, 6.81334075217897167063985453312, 7.31189777624683320020668248194, 8.121880471565918118942077558779, 9.094174067648897887992617595841, 10.12825230539877029429476143568