Properties

 Label 2-327-327.326-c0-0-2 Degree $2$ Conductor $327$ Sign $1$ Analytic cond. $0.163194$ Root an. cond. $0.403973$ Motivic weight $0$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

Origins

Dirichlet series

 L(s)  = 1 − 2-s + 3-s − 6-s − 7-s + 8-s + 9-s + 2·11-s + 14-s − 16-s − 17-s − 18-s − 21-s − 2·22-s − 23-s + 24-s + 25-s + 27-s − 31-s + 2·33-s + 34-s − 41-s + 42-s − 43-s + 46-s − 47-s − 48-s − 50-s + ⋯
 L(s)  = 1 − 2-s + 3-s − 6-s − 7-s + 8-s + 9-s + 2·11-s + 14-s − 16-s − 17-s − 18-s − 21-s − 2·22-s − 23-s + 24-s + 25-s + 27-s − 31-s + 2·33-s + 34-s − 41-s + 42-s − 43-s + 46-s − 47-s − 48-s − 50-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 327 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 327 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$327$$    =    $$3 \cdot 109$$ Sign: $1$ Analytic conductor: $$0.163194$$ Root analytic conductor: $$0.403973$$ Motivic weight: $$0$$ Rational: yes Arithmetic: yes Character: $\chi_{327} (326, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 327,\ (\ :0),\ 1)$$

Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.6173250271$$ $$L(\frac12)$$ $$\approx$$ $$0.6173250271$$ $$L(1)$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 - T$$
109 $$1 - T$$
good2 $$1 + T + T^{2}$$
5 $$( 1 - T )( 1 + T )$$
7 $$1 + T + T^{2}$$
11 $$( 1 - T )^{2}$$
13 $$( 1 - T )( 1 + T )$$
17 $$1 + T + T^{2}$$
19 $$( 1 - T )( 1 + T )$$
23 $$1 + T + T^{2}$$
29 $$( 1 - T )( 1 + T )$$
31 $$1 + T + T^{2}$$
37 $$( 1 - T )( 1 + T )$$
41 $$1 + T + T^{2}$$
43 $$1 + T + T^{2}$$
47 $$1 + T + T^{2}$$
53 $$( 1 - T )^{2}$$
59 $$1 + T + T^{2}$$
61 $$1 + T + T^{2}$$
67 $$( 1 - T )( 1 + T )$$
71 $$( 1 - T )( 1 + T )$$
73 $$1 + T + T^{2}$$
79 $$( 1 - T )( 1 + T )$$
83 $$( 1 - T )( 1 + T )$$
89 $$( 1 - T )( 1 + T )$$
97 $$1 + T + T^{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

−11.74650745927727950131795816314, −10.45945189387656472092812801547, −9.624193688403898187388115849847, −9.045224346074777022319737595946, −8.467812092544809563656608280323, −7.14944691097300723106608156832, −6.50100474412380231900694567502, −4.42386325006468296262737482580, −3.48379135769085281071487237739, −1.70458306709519005726191879779, 1.70458306709519005726191879779, 3.48379135769085281071487237739, 4.42386325006468296262737482580, 6.50100474412380231900694567502, 7.14944691097300723106608156832, 8.467812092544809563656608280323, 9.045224346074777022319737595946, 9.624193688403898187388115849847, 10.45945189387656472092812801547, 11.74650745927727950131795816314