Properties

 Label 2-42e2-1.1-c1-0-5 Degree $2$ Conductor $1764$ Sign $1$ Analytic cond. $14.0856$ Root an. cond. $3.75308$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 + 6·11-s − 2·13-s + 4·19-s + 6·23-s − 5·25-s − 6·29-s − 8·31-s + 2·37-s + 12·41-s − 4·43-s + 12·47-s + 6·53-s + 10·61-s + 8·67-s − 6·71-s + 10·73-s − 4·79-s − 12·83-s + 12·89-s + 10·97-s − 12·101-s − 8·103-s + 6·107-s + 14·109-s + 6·113-s + ⋯
 L(s)  = 1 + 1.80·11-s − 0.554·13-s + 0.917·19-s + 1.25·23-s − 25-s − 1.11·29-s − 1.43·31-s + 0.328·37-s + 1.87·41-s − 0.609·43-s + 1.75·47-s + 0.824·53-s + 1.28·61-s + 0.977·67-s − 0.712·71-s + 1.17·73-s − 0.450·79-s − 1.31·83-s + 1.27·89-s + 1.01·97-s − 1.19·101-s − 0.788·103-s + 0.580·107-s + 1.34·109-s + 0.564·113-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$1764$$    =    $$2^{2} \cdot 3^{2} \cdot 7^{2}$$ Sign: $1$ Analytic conductor: $$14.0856$$ Root analytic conductor: $$3.75308$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: $\chi_{1764} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 1764,\ (\ :1/2),\ 1)$$

Particular Values

 $$L(1)$$ $$\approx$$ $$1.911865514$$ $$L(\frac12)$$ $$\approx$$ $$1.911865514$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1$$
7 $$1$$
good5 $$1 + p T^{2}$$
11 $$1 - 6 T + p T^{2}$$
13 $$1 + 2 T + p T^{2}$$
17 $$1 + p T^{2}$$
19 $$1 - 4 T + p T^{2}$$
23 $$1 - 6 T + p T^{2}$$
29 $$1 + 6 T + p T^{2}$$
31 $$1 + 8 T + p T^{2}$$
37 $$1 - 2 T + p T^{2}$$
41 $$1 - 12 T + p T^{2}$$
43 $$1 + 4 T + p T^{2}$$
47 $$1 - 12 T + p T^{2}$$
53 $$1 - 6 T + p T^{2}$$
59 $$1 + p T^{2}$$
61 $$1 - 10 T + p T^{2}$$
67 $$1 - 8 T + p T^{2}$$
71 $$1 + 6 T + p T^{2}$$
73 $$1 - 10 T + p T^{2}$$
79 $$1 + 4 T + p T^{2}$$
83 $$1 + 12 T + p T^{2}$$
89 $$1 - 12 T + p T^{2}$$
97 $$1 - 10 T + p 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

−9.331187149331773045855656881837, −8.729935221736740713914818138949, −7.44499063873378776873294009662, −7.12758834797060137294534233139, −6.03492627650465326217694729948, −5.32133833421984252130695309604, −4.15927538579926872458234038883, −3.52771007221053761224054171247, −2.20338451459941845252791108478, −1.00219484121657416616236254344, 1.00219484121657416616236254344, 2.20338451459941845252791108478, 3.52771007221053761224054171247, 4.15927538579926872458234038883, 5.32133833421984252130695309604, 6.03492627650465326217694729948, 7.12758834797060137294534233139, 7.44499063873378776873294009662, 8.729935221736740713914818138949, 9.331187149331773045855656881837