Properties

 Degree $2$ Conductor $668$ Sign $1$ Motivic weight $1$ Primitive yes Self-dual yes Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 + 1.89·3-s + 2·5-s + 4.92·7-s + 0.601·9-s − 2.15·11-s − 3.32·13-s + 3.79·15-s + 5.91·17-s − 0.927·19-s + 9.33·21-s − 7.63·23-s − 25-s − 4.55·27-s − 1.59·29-s − 5.18·31-s − 4.09·33-s + 9.84·35-s + 1.53·37-s − 6.31·39-s − 9.71·41-s + 6.65·43-s + 1.20·45-s + 5.72·47-s + 17.2·49-s + 11.2·51-s − 9.24·53-s − 4.31·55-s + ⋯
 L(s)  = 1 + 1.09·3-s + 0.894·5-s + 1.85·7-s + 0.200·9-s − 0.649·11-s − 0.922·13-s + 0.979·15-s + 1.43·17-s − 0.212·19-s + 2.03·21-s − 1.59·23-s − 0.200·25-s − 0.876·27-s − 0.296·29-s − 0.930·31-s − 0.712·33-s + 1.66·35-s + 0.251·37-s − 1.01·39-s − 1.51·41-s + 1.01·43-s + 0.179·45-s + 0.834·47-s + 2.45·49-s + 1.57·51-s − 1.26·53-s − 0.581·55-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 668 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 668 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$668$$    =    $$2^{2} \cdot 167$$ Sign: $1$ Motivic weight: $$1$$ Character: $\chi_{668} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 668,\ (\ :1/2),\ 1)$$

Particular Values

 $$L(1)$$ $$\approx$$ $$2.66379$$ $$L(\frac12)$$ $$\approx$$ $$2.66379$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
167 $$1 + T$$
good3 $$1 - 1.89T + 3T^{2}$$
5 $$1 - 2T + 5T^{2}$$
7 $$1 - 4.92T + 7T^{2}$$
11 $$1 + 2.15T + 11T^{2}$$
13 $$1 + 3.32T + 13T^{2}$$
17 $$1 - 5.91T + 17T^{2}$$
19 $$1 + 0.927T + 19T^{2}$$
23 $$1 + 7.63T + 23T^{2}$$
29 $$1 + 1.59T + 29T^{2}$$
31 $$1 + 5.18T + 31T^{2}$$
37 $$1 - 1.53T + 37T^{2}$$
41 $$1 + 9.71T + 41T^{2}$$
43 $$1 - 6.65T + 43T^{2}$$
47 $$1 - 5.72T + 47T^{2}$$
53 $$1 + 9.24T + 53T^{2}$$
59 $$1 - 5.78T + 59T^{2}$$
61 $$1 - 14.8T + 61T^{2}$$
67 $$1 - 10.5T + 67T^{2}$$
71 $$1 - 5.51T + 71T^{2}$$
73 $$1 + 13.5T + 73T^{2}$$
79 $$1 + 8.23T + 79T^{2}$$
83 $$1 - 7.97T + 83T^{2}$$
89 $$1 + 8.41T + 89T^{2}$$
97 $$1 - 5.34T + 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

−10.25343884565562589118459553320, −9.700101264486565439160150642127, −8.625998352595403000151378837362, −7.938045256444632597035754417979, −7.46327189081337935861031128684, −5.72192644241367696862147939740, −5.15780096951604515523804045145, −3.87689014113227537308151753362, −2.42838613076373027708445734664, −1.79058045829787187667008636688, 1.79058045829787187667008636688, 2.42838613076373027708445734664, 3.87689014113227537308151753362, 5.15780096951604515523804045145, 5.72192644241367696862147939740, 7.46327189081337935861031128684, 7.938045256444632597035754417979, 8.625998352595403000151378837362, 9.700101264486565439160150642127, 10.25343884565562589118459553320