# Properties

 Degree 2 Conductor $2 \cdot 3 \cdot 11 \cdot 61$ Sign $1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + 2-s + 3-s + 4-s + 4.36·5-s + 6-s + 4.52·7-s + 8-s + 9-s + 4.36·10-s + 11-s + 12-s − 6.17·13-s + 4.52·14-s + 4.36·15-s + 16-s + 4.36·17-s + 18-s − 5.40·19-s + 4.36·20-s + 4.52·21-s + 22-s − 7.65·23-s + 24-s + 14.0·25-s − 6.17·26-s + 27-s + 4.52·28-s + ⋯
 L(s)  = 1 + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.95·5-s + 0.408·6-s + 1.70·7-s + 0.353·8-s + 0.333·9-s + 1.38·10-s + 0.301·11-s + 0.288·12-s − 1.71·13-s + 1.20·14-s + 1.12·15-s + 0.250·16-s + 1.05·17-s + 0.235·18-s − 1.23·19-s + 0.977·20-s + 0.986·21-s + 0.213·22-s − 1.59·23-s + 0.204·24-s + 2.81·25-s − 1.21·26-s + 0.192·27-s + 0.854·28-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$4026$$    =    $$2 \cdot 3 \cdot 11 \cdot 61$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{4026} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(2,\ 4026,\ (\ :1/2),\ 1)$ $L(1)$ $\approx$ $6.266382562$ $L(\frac12)$ $\approx$ $6.266382562$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;3,\;11,\;61\}$, $F_p(T) = 1 - a_p T + p T^2 .$If $p \in \{2,\;3,\;11,\;61\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 $$1 - T$$
3 $$1 - T$$
11 $$1 - T$$
61 $$1 + T$$
good5 $$1 - 4.36T + 5T^{2}$$
7 $$1 - 4.52T + 7T^{2}$$
13 $$1 + 6.17T + 13T^{2}$$
17 $$1 - 4.36T + 17T^{2}$$
19 $$1 + 5.40T + 19T^{2}$$
23 $$1 + 7.65T + 23T^{2}$$
29 $$1 + 7.78T + 29T^{2}$$
31 $$1 + 1.28T + 31T^{2}$$
37 $$1 + 0.611T + 37T^{2}$$
41 $$1 - 9.12T + 41T^{2}$$
43 $$1 + 9.01T + 43T^{2}$$
47 $$1 + 0.764T + 47T^{2}$$
53 $$1 + 0.479T + 53T^{2}$$
59 $$1 - 7.96T + 59T^{2}$$
67 $$1 + 6.16T + 67T^{2}$$
71 $$1 - 8.06T + 71T^{2}$$
73 $$1 + 6.83T + 73T^{2}$$
79 $$1 - 2.99T + 79T^{2}$$
83 $$1 + 12.1T + 83T^{2}$$
89 $$1 - 10.2T + 89T^{2}$$
97 $$1 + 12.3T + 97T^{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

−8.360440952730460630674916011537, −7.68743845558300489657235041409, −6.95498778500302412242871978155, −5.96792006205516497523963831101, −5.41726746283321060898078072743, −4.80795139012144639589608708276, −4.01175268049694801052942102681, −2.62887761785164225856460799792, −2.02561692451421263865853388123, −1.57614262881653932199022393951, 1.57614262881653932199022393951, 2.02561692451421263865853388123, 2.62887761785164225856460799792, 4.01175268049694801052942102681, 4.80795139012144639589608708276, 5.41726746283321060898078072743, 5.96792006205516497523963831101, 6.95498778500302412242871978155, 7.68743845558300489657235041409, 8.360440952730460630674916011537