Properties

 Label 2-7e2-1.1-c1-0-0 Degree $2$ Conductor $49$ Sign $1$ Analytic cond. $0.391266$ Root an. cond. $0.625513$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

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Dirichlet series

 L(s)  = 1 + 2-s − 4-s − 3·8-s − 3·9-s + 4·11-s − 16-s − 3·18-s + 4·22-s + 8·23-s − 5·25-s + 2·29-s + 5·32-s + 3·36-s − 6·37-s − 12·43-s − 4·44-s + 8·46-s − 5·50-s − 10·53-s + 2·58-s + 7·64-s + 4·67-s + 16·71-s + 9·72-s − 6·74-s + 8·79-s + 9·81-s + ⋯
 L(s)  = 1 + 0.707·2-s − 1/2·4-s − 1.06·8-s − 9-s + 1.20·11-s − 1/4·16-s − 0.707·18-s + 0.852·22-s + 1.66·23-s − 25-s + 0.371·29-s + 0.883·32-s + 1/2·36-s − 0.986·37-s − 1.82·43-s − 0.603·44-s + 1.17·46-s − 0.707·50-s − 1.37·53-s + 0.262·58-s + 7/8·64-s + 0.488·67-s + 1.89·71-s + 1.06·72-s − 0.697·74-s + 0.900·79-s + 81-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$49$$    =    $$7^{2}$$ Sign: $1$ Analytic conductor: $$0.391266$$ Root analytic conductor: $$0.625513$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 49,\ (\ :1/2),\ 1)$$

Particular Values

 $$L(1)$$ $$\approx$$ $$0.9666558528$$ $$L(\frac12)$$ $$\approx$$ $$0.9666558528$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad7 $$1$$
good2 $$1 - T + p T^{2}$$
3 $$1 + p T^{2}$$
5 $$1 + p T^{2}$$
11 $$1 - 4 T + p T^{2}$$
13 $$1 + p T^{2}$$
17 $$1 + p T^{2}$$
19 $$1 + p T^{2}$$
23 $$1 - 8 T + p T^{2}$$
29 $$1 - 2 T + p T^{2}$$
31 $$1 + p T^{2}$$
37 $$1 + 6 T + p T^{2}$$
41 $$1 + p T^{2}$$
43 $$1 + 12 T + p T^{2}$$
47 $$1 + p T^{2}$$
53 $$1 + 10 T + p T^{2}$$
59 $$1 + p T^{2}$$
61 $$1 + p T^{2}$$
67 $$1 - 4 T + p T^{2}$$
71 $$1 - 16 T + p T^{2}$$
73 $$1 + p T^{2}$$
79 $$1 - 8 T + p T^{2}$$
83 $$1 + p T^{2}$$
89 $$1 + p T^{2}$$
97 $$1 + 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

−15.25649719028727745654998030256, −14.36789003254179224985767994212, −13.55829126515041538892762617601, −12.27943344760252552732176729088, −11.30850265031875128052013945557, −9.489525085039792097687572288594, −8.498120181782134288657898916471, −6.47803659589426412986704519993, −5.08673463814783736975452386140, −3.45773984941604093394269964405, 3.45773984941604093394269964405, 5.08673463814783736975452386140, 6.47803659589426412986704519993, 8.498120181782134288657898916471, 9.489525085039792097687572288594, 11.30850265031875128052013945557, 12.27943344760252552732176729088, 13.55829126515041538892762617601, 14.36789003254179224985767994212, 15.25649719028727745654998030256