# Properties

 Label 2-6897-1.1-c1-0-259 Degree $2$ Conductor $6897$ Sign $-1$ Analytic cond. $55.0728$ Root an. cond. $7.42110$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $1$

# Origins

## Dirichlet series

 L(s)  = 1 + 3-s − 2·4-s + 2·5-s + 9-s − 2·12-s − 5·13-s + 2·15-s + 4·16-s − 5·17-s + 19-s − 4·20-s − 4·23-s − 25-s + 27-s + 10·29-s + 10·31-s − 2·36-s + 2·37-s − 5·39-s − 10·43-s + 2·45-s + 8·47-s + 4·48-s − 7·49-s − 5·51-s + 10·52-s + 53-s + ⋯
 L(s)  = 1 + 0.577·3-s − 4-s + 0.894·5-s + 1/3·9-s − 0.577·12-s − 1.38·13-s + 0.516·15-s + 16-s − 1.21·17-s + 0.229·19-s − 0.894·20-s − 0.834·23-s − 1/5·25-s + 0.192·27-s + 1.85·29-s + 1.79·31-s − 1/3·36-s + 0.328·37-s − 0.800·39-s − 1.52·43-s + 0.298·45-s + 1.16·47-s + 0.577·48-s − 49-s − 0.700·51-s + 1.38·52-s + 0.137·53-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 6897 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 6897 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$6897$$    =    $$3 \cdot 11^{2} \cdot 19$$ Sign: $-1$ Analytic conductor: $$55.0728$$ Root analytic conductor: $$7.42110$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 6897,\ (\ :1/2),\ -1)$$

## Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 - T$$
11 $$1$$
19 $$1 - T$$
good2 $$1 + p T^{2}$$
5 $$1 - 2 T + p T^{2}$$
7 $$1 + p T^{2}$$
13 $$1 + 5 T + p T^{2}$$
17 $$1 + 5 T + p T^{2}$$
23 $$1 + 4 T + p T^{2}$$
29 $$1 - 10 T + p T^{2}$$
31 $$1 - 10 T + p T^{2}$$
37 $$1 - 2 T + p T^{2}$$
41 $$1 + p T^{2}$$
43 $$1 + 10 T + p T^{2}$$
47 $$1 - 8 T + p T^{2}$$
53 $$1 - T + p T^{2}$$
59 $$1 + 5 T + p T^{2}$$
61 $$1 + 10 T + p T^{2}$$
67 $$1 - 8 T + p T^{2}$$
71 $$1 + 3 T + p T^{2}$$
73 $$1 + 10 T + p T^{2}$$
79 $$1 - 15 T + p T^{2}$$
83 $$1 + 15 T + p T^{2}$$
89 $$1 + 9 T + p T^{2}$$
97 $$1 + 8 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

−7.893975174648438801263745489809, −6.82777667960537683851191134491, −6.27297975844116056273583281523, −5.33842116158395541106055859262, −4.62977534652502950638760594586, −4.22797059300465569011259269641, −2.97122527355802937375186698151, −2.41424876162209710552864646413, −1.36172689789925662802106600482, 0, 1.36172689789925662802106600482, 2.41424876162209710552864646413, 2.97122527355802937375186698151, 4.22797059300465569011259269641, 4.62977534652502950638760594586, 5.33842116158395541106055859262, 6.27297975844116056273583281523, 6.82777667960537683851191134491, 7.893975174648438801263745489809