# Properties

 Label 2-7e2-1.1-c3-0-2 Degree $2$ Conductor $49$ Sign $-1$ Analytic cond. $2.89109$ Root an. cond. $1.70032$ Motivic weight $3$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $1$

# Origins

## Dirichlet series

 L(s)  = 1 − 5·2-s + 17·4-s − 45·8-s − 27·9-s − 68·11-s + 89·16-s + 135·18-s + 340·22-s − 40·23-s − 125·25-s − 166·29-s − 85·32-s − 459·36-s + 450·37-s − 180·43-s − 1.15e3·44-s + 200·46-s + 625·50-s + 590·53-s + 830·58-s − 287·64-s − 740·67-s + 688·71-s + 1.21e3·72-s − 2.25e3·74-s − 1.38e3·79-s + 729·81-s + ⋯
 L(s)  = 1 − 1.76·2-s + 17/8·4-s − 1.98·8-s − 9-s − 1.86·11-s + 1.39·16-s + 1.76·18-s + 3.29·22-s − 0.362·23-s − 25-s − 1.06·29-s − 0.469·32-s − 2.12·36-s + 1.99·37-s − 0.638·43-s − 3.96·44-s + 0.641·46-s + 1.76·50-s + 1.52·53-s + 1.87·58-s − 0.560·64-s − 1.34·67-s + 1.15·71-s + 1.98·72-s − 3.53·74-s − 1.97·79-s + 81-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$49$$    =    $$7^{2}$$ Sign: $-1$ Analytic conductor: $$2.89109$$ Root analytic conductor: $$1.70032$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: $\chi_{49} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 49,\ (\ :3/2),\ -1)$$

## Particular Values

 $$L(2)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{5}{2})$$ 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 + 5 T + p^{3} T^{2}$$
3 $$1 + p^{3} T^{2}$$
5 $$1 + p^{3} T^{2}$$
11 $$1 + 68 T + p^{3} T^{2}$$
13 $$1 + p^{3} T^{2}$$
17 $$1 + p^{3} T^{2}$$
19 $$1 + p^{3} T^{2}$$
23 $$1 + 40 T + p^{3} T^{2}$$
29 $$1 + 166 T + p^{3} T^{2}$$
31 $$1 + p^{3} T^{2}$$
37 $$1 - 450 T + p^{3} T^{2}$$
41 $$1 + p^{3} T^{2}$$
43 $$1 + 180 T + p^{3} T^{2}$$
47 $$1 + p^{3} T^{2}$$
53 $$1 - 590 T + p^{3} T^{2}$$
59 $$1 + p^{3} T^{2}$$
61 $$1 + p^{3} T^{2}$$
67 $$1 + 740 T + p^{3} T^{2}$$
71 $$1 - 688 T + p^{3} T^{2}$$
73 $$1 + p^{3} T^{2}$$
79 $$1 + 1384 T + p^{3} T^{2}$$
83 $$1 + p^{3} T^{2}$$
89 $$1 + p^{3} T^{2}$$
97 $$1 + p^{3} 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

−15.02894184714965737715717038841, −13.30442063818121698660187280692, −11.64156989511880964719127413578, −10.69837535176370904724193150519, −9.657416614835928522773169291177, −8.353921336728576853261425161019, −7.56866168292611020750437653852, −5.77862461637823030407687672667, −2.50258280312093899836671041842, 0, 2.50258280312093899836671041842, 5.77862461637823030407687672667, 7.56866168292611020750437653852, 8.353921336728576853261425161019, 9.657416614835928522773169291177, 10.69837535176370904724193150519, 11.64156989511880964719127413578, 13.30442063818121698660187280692, 15.02894184714965737715717038841