# Properties

 Label 2-35e2-1.1-c3-0-178 Degree $2$ Conductor $1225$ Sign $-1$ Analytic cond. $72.2773$ Root an. cond. $8.50160$ 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 − 166·29-s + 85·32-s − 459·36-s − 450·37-s + 180·43-s − 1.15e3·44-s + 200·46-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 + 900·86-s − 3.06e3·88-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 − 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 − 1.52·53-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 + 1.12·86-s − 3.70·88-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1225$$    =    $$5^{2} \cdot 7^{2}$$ Sign: $-1$ Analytic conductor: $$72.2773$$ Root analytic conductor: $$8.50160$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: $\chi_{1225} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 1225,\ (\ :3/2),\ -1)$$

## 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$$F_p(T)$
bad5 $$1$$
7 $$1$$
good2 $$1 - 5 T + p^{3} T^{2}$$
3 $$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

−8.784853078304363255888346571584, −7.85042915357328648786477403234, −7.08012648942760879408187945070, −6.03074505577547228334097971392, −5.35839426906832944408285085560, −4.87919502475217459673920068436, −3.60814427670431172414485522132, −2.88453729935879048586939345379, −2.06407466010492228484826349351, 0, 2.06407466010492228484826349351, 2.88453729935879048586939345379, 3.60814427670431172414485522132, 4.87919502475217459673920068436, 5.35839426906832944408285085560, 6.03074505577547228334097971392, 7.08012648942760879408187945070, 7.85042915357328648786477403234, 8.784853078304363255888346571584