# Properties

 Degree 2 Conductor $7 \cdot 157$ Sign $-1$ Motivic weight 0 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 − 1.41i·2-s − 1.41i·3-s − 1.00·4-s + 5-s − 2.00·6-s + 7-s − 1.00·9-s − 1.41i·10-s − 11-s + 1.41i·12-s + 1.41i·13-s − 1.41i·14-s − 1.41i·15-s − 0.999·16-s + 1.41i·17-s + 1.41i·18-s + ⋯
 L(s)  = 1 − 1.41i·2-s − 1.41i·3-s − 1.00·4-s + 5-s − 2.00·6-s + 7-s − 1.00·9-s − 1.41i·10-s − 11-s + 1.41i·12-s + 1.41i·13-s − 1.41i·14-s − 1.41i·15-s − 0.999·16-s + 1.41i·17-s + 1.41i·18-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1099 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1099 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$1099$$    =    $$7 \cdot 157$$ $$\varepsilon$$ = $-1$ motivic weight = $$0$$ character : $\chi_{1099} (1098, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 1099,\ (\ :0),\ -1)$$ $$L(\frac{1}{2})$$ $$\approx$$ $$1.255061004$$ $$L(\frac12)$$ $$\approx$$ $$1.255061004$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{7,\;157\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{7,\;157\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad7 $$1 - T$$
157 $$1 + T$$
good2 $$1 + 1.41iT - T^{2}$$
3 $$1 + 1.41iT - T^{2}$$
5 $$1 - T + T^{2}$$
11 $$1 + T + T^{2}$$
13 $$1 - 1.41iT - T^{2}$$
17 $$1 - 1.41iT - T^{2}$$
19 $$1 + 1.41iT - T^{2}$$
23 $$1 + 1.41iT - T^{2}$$
29 $$1 - T^{2}$$
31 $$1 - 1.41iT - T^{2}$$
37 $$1 + T + T^{2}$$
41 $$1 + T^{2}$$
43 $$1 - 1.41iT - T^{2}$$
47 $$1 - T^{2}$$
53 $$1 - T^{2}$$
59 $$1 - T + T^{2}$$
61 $$1 - T + T^{2}$$
67 $$1 - T + T^{2}$$
71 $$1 + T + T^{2}$$
73 $$1 - T + T^{2}$$
79 $$1 - 1.41iT - T^{2}$$
83 $$1 + T^{2}$$
89 $$1 - T^{2}$$
97 $$1 + T + T^{2}$$
show less
\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

−9.888377561001886859564733883998, −8.808563190495100538701679310691, −8.251207927507114370158544227628, −6.96812051134157432931337368838, −6.50157303852040334319083637926, −5.22311812951748254059338341638, −4.25961002387416376719613084651, −2.60809052647635850507664761521, −2.04019478288948419773736791401, −1.32775542764724275912616004579, 2.28873323938685465760522650434, 3.69409142897207908743715002290, 5.00137227226463542801986455885, 5.41150114349308182059975919530, 5.77791151515662225600748266041, 7.34144124055477092027925249526, 7.904838050419210420809007195296, 8.719103325454428039134100272174, 9.684516409772640015859514790217, 10.14708363041210456982881773240