# Properties

 Degree 2 Conductor $2^{2}$ Sign $1$ Motivic weight 17 Primitive yes Self-dual yes Analytic rank 0

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## Dirichlet series

 L(s)  = 1 − 2.15e4·3-s + 9.70e5·5-s + 1.64e6·7-s + 3.33e8·9-s − 7.82e7·11-s + 2.36e9·13-s − 2.08e10·15-s − 1.04e10·17-s + 1.03e11·19-s − 3.53e10·21-s + 3.30e11·23-s + 1.79e11·25-s − 4.40e12·27-s + 3.69e12·29-s + 2.53e12·31-s + 1.68e12·33-s + 1.59e12·35-s + 1.96e13·37-s − 5.08e13·39-s − 3.32e13·41-s − 8.49e13·43-s + 3.24e14·45-s + 2.18e12·47-s − 2.29e14·49-s + 2.25e14·51-s − 1.81e13·53-s − 7.59e13·55-s + ⋯
 L(s)  = 1 − 1.89·3-s + 1.11·5-s + 0.107·7-s + 2.58·9-s − 0.110·11-s + 0.803·13-s − 2.10·15-s − 0.364·17-s + 1.40·19-s − 0.203·21-s + 0.881·23-s + 0.235·25-s − 3.00·27-s + 1.37·29-s + 0.533·31-s + 0.208·33-s + 0.119·35-s + 0.919·37-s − 1.52·39-s − 0.650·41-s − 1.10·43-s + 2.87·45-s + 0.0133·47-s − 0.988·49-s + 0.691·51-s − 0.0401·53-s − 0.122·55-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$4$$    =    $$2^{2}$$ $$\varepsilon$$ = $1$ motivic weight = $$17$$ character : $\chi_{4} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(2,\ 4,\ (\ :17/2),\ 1)$ $L(9)$ $\approx$ $1.13846$ $L(\frac12)$ $\approx$ $1.13846$ $L(\frac{19}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 2$, $$F_p(T)$$ is a polynomial of degree 2. If $p = 2$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1$$
good3 $$1 + 2.15e4T + 1.29e8T^{2}$$
5 $$1 - 9.70e5T + 7.62e11T^{2}$$
7 $$1 - 1.64e6T + 2.32e14T^{2}$$
11 $$1 + 7.82e7T + 5.05e17T^{2}$$
13 $$1 - 2.36e9T + 8.65e18T^{2}$$
17 $$1 + 1.04e10T + 8.27e20T^{2}$$
19 $$1 - 1.03e11T + 5.48e21T^{2}$$
23 $$1 - 3.30e11T + 1.41e23T^{2}$$
29 $$1 - 3.69e12T + 7.25e24T^{2}$$
31 $$1 - 2.53e12T + 2.25e25T^{2}$$
37 $$1 - 1.96e13T + 4.56e26T^{2}$$
41 $$1 + 3.32e13T + 2.61e27T^{2}$$
43 $$1 + 8.49e13T + 5.87e27T^{2}$$
47 $$1 - 2.18e12T + 2.66e28T^{2}$$
53 $$1 + 1.81e13T + 2.05e29T^{2}$$
59 $$1 - 1.24e15T + 1.27e30T^{2}$$
61 $$1 - 2.09e15T + 2.24e30T^{2}$$
67 $$1 + 2.53e15T + 1.10e31T^{2}$$
71 $$1 - 8.39e15T + 2.96e31T^{2}$$
73 $$1 + 6.50e15T + 4.74e31T^{2}$$
79 $$1 + 1.41e16T + 1.81e32T^{2}$$
83 $$1 - 1.76e16T + 4.21e32T^{2}$$
89 $$1 - 2.17e16T + 1.37e33T^{2}$$
97 $$1 + 8.31e16T + 5.95e33T^{2}$$
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\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}

## Imaginary part of the first few zeros on the critical line

−21.17200737763757841361731633284, −18.27783764260195604388851924085, −17.42272531046447011513111920059, −16.04523214199944693054209519596, −13.28890716299919749587649571602, −11.52409998933883939093132771380, −10.04523180569335592908663683829, −6.48035812582763516333469305742, −5.16076977806956282797043243787, −1.11495662939605257989208254741, 1.11495662939605257989208254741, 5.16076977806956282797043243787, 6.48035812582763516333469305742, 10.04523180569335592908663683829, 11.52409998933883939093132771380, 13.28890716299919749587649571602, 16.04523214199944693054209519596, 17.42272531046447011513111920059, 18.27783764260195604388851924085, 21.17200737763757841361731633284